pdsolve(a_*b_,y_):=pdsolve(a,y) and pdsolve(b,y); #2; pdsolve(y(p_,t_)*y(q_,x_)+d_*y_^2,y_):= if(isconstant(d), C_1*exp((-d)^(1/2/p)*t+(-d)^(1/2/q)*x) ); pdsolve(y(p_,t_)*y(q_,x_)+d_*(y_+c_)^2,y_):= if(isconstant(c,d), -c+C_1*exp((-d)^(1/2/p)*t+(-d)^(1/2/q)*x) ); pdsolve(y(p_,t_)*y(q_,x_)+d_*(b_*y_+c_)^2,y_):= if(isconstant(b,c,d), -c/b+C_1*exp(b^(1/p)*(-d)^(1/2/p)*t+b^(1/q)*(-d)^(1/2/q)*x) ); pdsolve(y(1,t_)^n_+y_^m_*y(1,x_)^n_,y_):=(-1)^(1/m+n)*(C_1+m*x)^(n/m)/(C_2+m*t)^(n/m); pdsolve(y(1,t_)^n_+y_*y(1,x_)^n_,y_):=(-1)^(n+1)*(C_1+x)^n/(C_2+t)^n; pdsolve(y(p_,t_)+b_*y(q_,x_)^n_,y_):= if(isconstant(b),C_1*(-b*t^p/p!+x^q/q!)+gsolution(0,y,t,p)+gsolution(0,y,x,q) ); pdsolve(y(p_,t_)+y(q_,x_)^n_,y_):= C_1*(-t^p/p!+x^q/q!)+gsolution(0,y,t,p)+gsolution(0,y,x,q); pdsolve(y(p_,t_)+y(q_,x_),y_):= C_1*(t^p/p!-x^q/q!)+if(p>1 and q>1, C_2*t*x,0)+gsolution(0,y,t,p)+gsolution(0,y,x,q); pdsolve1(y(p_,t_)+b_*y(q_,x_),y_):= if(hasnot(b,y), C_1*(if(hasnot(b,t),if(hasnot(b,x),t^p/p!-x^q/q!/b, t^p/p!-int(1/b,x,q)), int(b,t,p)-x^q/q!)) +if(p>1 and q>1, C_2*t*x,0)+gsolution(0,y,t,p)+gsolution(0,y,x,q) ); pdsolve(a_*y(p_,t_)+y(q_,x_),y_):= if(hasnot(a,t), C_1*(t^p/p!-int(a,x,q)),0) +if(hasnot(a,x), C_1*(int(1/a,t,p)-x^q/q!),0) +if(p>1 and q>1, C_2*t*x,0)+gsolution(0,y,t,p)+gsolution(0,y,x,q); pdsolve(a_*y(p_,t_)+b_*y(q_,x_),y_):= if(hasnot(a,t) and hasnot(b,x), C_1*(int(b,t,p)-int(a,x,q)), if(hasnot(a,x) and hasnot(b,t), C_1*(int(1/a,t,p)-int(1/b,x,q)) )) +if(p>1 and q>1, C_2*t*x,0)+gsolution(0,y,t,p)+gsolution(0,y,x,q); #pdsolve(y(p_,t_)+a_*y(2,x_),y_):= if(hasnot(a,y), if(hasnot(a,x) and p<=1, C_1+C_2*(t^p/p!+x-x^2/2/a)+C_3*E(p,a*t^p)*(sin(x)+cos(x)), C_1+C_2*x+int(-1/a,x,2)+t^p/p! )); #pdsolve(y(2,t_)+b_*y(2,x_),y_):= if(isconstant(b), C_1+C_3*(t+x+t^2-x^2/b)+C_2*(sin(x)+cos(x))*if(b>0,(exp(-sqrt(b)*t)+exp(sqrt(b)*t)),(sin(sqrt(-b)*t)+cos(sqrt(-b)*t)) ) ); #pdsolve(y(1,t_)+a_*y(1,x_),y_):= if(hasnot(a,y), if(hasnot(a,x), C_1+C_2*(t-x/a), C_1+t-int(1/a,x) )); #pdsolve(y(1,t_)+b_*y(1,x_),y_):= if(hasnot(b,y), if(hasnot(b,x), C_1+C_2*(t-x/b), C_1+t-int(1/b,x) ), if(hasnot(b,t) and hasnot(b,x),solve(b=(c_1+x)/(c_2+t),y) )); #pdsolve(a_*y(1,t_)+y(1,x_),y_):= if(has(a,y), if(hasnot(a,x),solve(a=(C_1+t)/(C_2+x),y), if(hasnot(a,t),solve(a=(C_1+t)/x,y) )),C_1+C_2*(x-int(1/a,t)) ); #pdsolve(a_*y(1,t_)+b_*y(1,x_),y_):= if(hasnot(a,t) and hasnot(b,x), solve(int(a,x)+int(b,t)+d(b,t)/d(b,x),y), if(hasnot(a,x) and hasnot(b,t), C_1*(int(1/a,t)-int(1/b,x)) )); pdsolve(y(1,t_)+b_*y(1,x_),y_):= if(has(b,y), if(has(b,x),if(has(b,t), b=-d(b,t)/d(b,x)), b=(C_1+x)/(C_2+t) ), if(hasnot(b,t),C_1+C_2*(t-int(1/b,x)), if(hasnot(b,x),C_1+C_2*(x-int(b,t)) ))); pdsolve(y(p_,t_)+y_^n_*y(q_,x_),y_):= -(C_1+t)^(-p/n)*(C_2+x)^(q/n)*fallingfactorial(-p/n,p)/fallingfactorial(q/n,q); pdsolve(y(p_,t_)+y_*y(q_,x_),y_):= -(C_1+t)^(-p)*(x^q+gsolution(0,y,x,q))*fallingfactorial(-p,p)/q!; pdsolve(y(p_,t_)+(b_+y_)*y(q_,x_),y_):= if(d(b,t,p)==0 and d(b,x,q)==0, -b-fallingfactorial(-p,p)*(C_1+t)^(-p)*(x^q+gsolution(0,y,x,q))/q!); pdsolve(y(p_,t_)+(b_+c_*y_)*y(q_,x_),y_):= if(isconstant(c) and d(b,t,p)==0 and d(b,x,q)==0, -b/c-1/c*fallingfactorial(-p,p)*(C_1+t)^(-p)*(x^q+gsolution(0,y,x,q))/q! ); pdsolve(y(p_,t_)+exp(a_+b_)*y(q_,x_),y_):= if(hasnot(a,x) and hasnot(b,y),C_1*(int(exp(a),t,p)-int(exp(-b),x,q)) ); pdsolve1(y(p_,t_)+a_*b_*y(q_,x_),y_):= if(hasnot(a*b,y),if(hasnot(b,t) and hasnot(b,x), C_1*(a*b*t^p/p!-x^q/q!)+if(p>1 and q>1,C_2*t*x,0),0) +if(has(b,x), C_1*(int(a,t,p)-int(1/b,x,q)),0) +if(has(b,t), C_1*(int(b,t,p)-int(1/a,x,q)),0) +gsolution(0,y,t,p)+gsolution(0,y,x,q) ); #pdsolve(y(p_,t_)+b_*y_^n_*y(q_,x_),y_):= if(hasnot(b,t),(-1)^(1/n)*(C_1+fallingfactorial(-p/n,p)^(-1/p)*t)^(-p/n)*(C_2+fallingfactorial(q/n,q)^(-1/q)*int(1/b,x,q))^(q/n) ); pdsolve1(y(p_,t_)+b_*y_*y(q_,x_),y_):= if(hasnot(b,x), 1/(C_1+int(b,t,p))*(gsolution(0,y,x,q)+x^q/q!), if(hasnot(b,t), 1/(C_1+factorial(p)*t^p)*(gsolution(0,y,x,q)+int(1/b,x,q)) )); pdsolve1(y(p_,t_)+b_*y_^n_*y(q_,x_),y_):= if(hasnot(b,x), (1/(C_1+int(b,t,p))*(gsolution(0,y,x,q)+x^q/q!))^(1/n), if(hasnot(b,t), (1/(C_1+factorial(p)*t^p)*(gsolution(0,y,x,q)+int(1/b,x,q)))^(1/n) )); pdsolve(y(p_,t_)+a_*b_*y(q_,x_),y_):= if(p<=1 and q<=1, if(has(b,y), if(has(a,t),solve(b=(C_1+x)/(C_2+int(a,t)),y), if(has(a,x),solve(b=(C_1+int(1/a,x))/(C_2+t),y) )), if(has(a,y), if(has(b,t),solve(a=(C_1+x)/(C_2+int(b,t)),y), if(has(b,x),solve(a=(C_1+int(1/b,x))/(C_2+t),y) ))))); pdsolve(y(p_,t_)+a_*b_*c_*y(q_,x_),y_):= if(p<=1 and q<=1, if(has(c,y), if(hasnot(a,x) and hasnot(b,t), c=(int(1/b,x,q)+C_2)/(C_1+int(a,t,p)), if(hasnot(a,t) and hasnot(b,x), c=((int(1/a,x,q)+C_2)/(C_1+int(b,t,p))) )), if(has(b,y), if(hasnot(a,x) and hasnot(c,t),solve(b=(C_1+int(1/c,x,q))/(C_2+int(a,t)),y), if(hasnot(a,t) and hasnot(c,x),solve(b=(C_1+int(1/a,x))/(C_2+int(c,t,p)),y) )), if(has(a,y), if(hasnot(b,x) and hasnot(c,t),solve(a=(C_1+int(1/c,x,q))/(C_2+int(b,t)),y), if(hasnot(b,t) and hasnot(c,x),solve(a=(C_1+int(1/b,x))/(C_2+int(c,t,p)),y) )))))); #pdsolve(y(p_,t_)+a_*b_*y_^n_*y(q_,x_),y_):= if(p<=1 and q<=1 and hasnot(a,b,y), if(hasnot(a,x) and hasnot(b,t), ((int(1/b,x,q)+C_2)/(C_1+int(a,t,p)))^(1/n), if(hasnot(a,t) and hasnot(b,x), ((int(1/a,x,q)+C_2)/(C_1+int(b,t,p)))^(1/n) ))); #pdsolve(y(p_,t_)+a_*b_*y_*y(q_,x_),y_):= if(p<=1 and q<=1 and hasnot(a,b,y), if(hasnot(a,x) and hasnot(b,t), (int(1/b,x,q)+C_2)/(C_1+int(a,t,p)), if(hasnot(a,t) and hasnot(b,x), (int(1/a,x,q)+C_2)/(C_1+int(b,t,p)) ))); pdsolve(y(1,t_)+a_*y(2,x_),y_):= if(has(a,y) or if(isfunction(a),d(argument(a,2),x)==0), int(1/int(1/a,y),y)=C_1-t+x and int(1/int(1/a,y),y)=C_1-t-x ); #pdsolve(y(1,t_)+a_*b_*y(2,x_),y_):= if(has(b,y) or if(isfunction(b),d(argument(b,2),x)==0),int(1/int(1/b,y),y)=C_1-int(a,t)+x, if(has(a,y) or if(isfunction(a),d(argument(a,2),x)==0),int(1/int(1/a,y),y)=C_1-int(b,t)+x )); pdsolve(y(1,t_)+a_*b_*y(2,x_),y_):= if(has(b,y),int(1/int(1/b,y),y)=C_1-int(a,t)+x, if(has(a,y),int(1/int(1/a,y),y)=C_1-int(b,t)+x )); pdsolve(y(p_,t_)+c_*exp(a_+b_)*y(q_,x_),y_):= if(hasnot(a,c,x) and hasnot(b,y),C_1*(int(c*exp(a),t,p)-int(exp(-b),x,q)) ); pdsolve(y(p_,t_)+exp(a_+b_)*f_*y(q_,x_),y_):= if(hasnot(a,f,x) and hasnot(b,y),C_1*(int(f*exp(a),t,p)-int(exp(-b),x,q)) ); pdsolve(y(p_,t_)+exp(c_+y_)*y(q_,x_),y_):=if(hasnot(c,t), log((-1)^(p+q)*p!*int(exp(-c),x,q))-p*log((C_2+t)), if(hasnot(c,x), log(-x^q/q!)-log(C_2+int(exp(c),t,p)) )); pdsolve1(y(p_,t_)+exp(y_)*f_*y(q_,x_),y_):= if(hasnot(f,x), -log(C_1+int(f,t,p))+log((-1)^(p+q)*p!/q!*x^q) ); pdsolve1(y(p_,t_)+b_*exp(y_)*y(q_,x_),y_):= if(hasnot(b,x), -log(C_1+int(b,t,p))+log((-1)^(p+q)*p!/q!*x^q) ); pdsolve(y(p_,t_)+exp(a_+b_)*y_*y(q_,x_),y_):= if(hasnot(a,x) and hasnot(b,t), (-1)^p/int(exp(a),t,p)*(gsolution(0,y,x,q)-int(exp(-b),x,q)) ); pdsolve(y(p_,t_)+exp(y_)*y(q_,x_),y_):= -p*log(C_1+t)+log(if(p+q>1 and p<>q,(-1)^(p+q),1)*p!/q!*x^q); pdsolve(y(p_,t_)+c_*exp(a_+b_)*y_*y(q_,x_),y_):= if(hasnot(a,c,x) and hasnot(b,c,t), (-1)^p*(gsolution(0,y,x,q)-int(exp(-b),x,q))/int(exp(a),t,p)/c ); pdsolve(y(1,t_)+(x_+b_)*y(1,x_),y_):= C_1*(exp(-t)*x-int(exp(-t)*b,t))+C_2; pdsolve(y(1,t_)+(x_+t_)*y(1,x_),y_):= C_1*exp(-t)*(t+x+1)+C_2; pdsolve(y(1,t_)+(a_*x_+b_)*y(1,x_),y_):= C_1*(exp(-int(a,t))*x-int(exp(-int(a,t))*b,t))+C_2; pdsolve(y(1,t_)+(a_*t_+b_*x_)*y(1,x_),y_):= C_1*(replace(int(1/(a+b*v),v)-t,v,a*t+b*x))+C_2; pdsolve(y(1,t_)+(a_*t_+b_*x_+c_)*y(1,x_),y_):= C_1*(replace(int(1/(a+b*v),v)-t,v,a*t+b*x+c))+C_2; #3; #pdsolve(y(p_,t_)+a_*y(q_,x_)+d_,y_):= if(hasnot(d,y), if(a<0, C_3*exp(t+x/(-a)^(1/q)),0)+C_1+C_2*(t^p/p!-int(1/a,x,q)) -if(has(d,t),if(has(d,x),if(isinteger(q), psolution(-a,1,0,d,y,t,x,p), psolution(0,d,p,x,q)/a), int(d,t,p) ), int(d/a,x,q) )); #pdsolve(y(1,t)+a_*y(1,x)+d_,y_):= if(has(d,x) and hasnot(d,t), C_1+C_2*F(t-x/a)+expand(int(replace(-d,t,t-x/a+z/a)*replace(-d,x,z),z,0,x)/a) ); #pdsolve(y(q_,t_)+b_*y(q_,x_)+c_,y_):= if(hasnot(b,c,t) and hasnot(b,c,x),if(b==-1, replace(dsolve(c,y,x,q),c_1,c_1*(t^q+x^q)), block(f:=dsolve(-c/(1+b),y,x,q),if(has(f,x^q), replace(f,x^q,t^q+x^q),replace(f,x,t+x))) )); #pdsolve(a_*y(q_,t_)+b_*y(q_,x_)+c_,y_):= if(hasnot(a,b,c,t) and hasnot(a,b,c,x), block(f:=dsolve(-c/(a+b),y,x,q),if(has(f,x^q), replace(f,x^q,t^q+x^q),replace(f,x,t+x))) ); #pdsolve(y(q_,t_)+y(q_,x_)+c_,y_):= if(hasnot(c,t) and hasnot(c,x), block(f:=dsolve(-c/2,y,x,q),if(has(f,x^q), replace(f,x^q,t^q+x^q),replace(f,x,t+x))) ); #pdsolve(y(1,t_)+y(1,x_)+f_,y_):= if(has(f,y),if(has(f,x),if(has(f,t), solve(f-d(argument(f,1),t)-d(argument(f,1),x),y), replace(replace(dsolve(-f,y,x,1),C_1,C_1+C_2*(t-x)),c_1,C_1+C_2*(t-x)) ))); #pdsolve(y(1,t_)+b_*y(1,x_)+f_,y_):= if(has(f,y),if(has(f,x),if(has(f,t), solve(f-d(argument(f,1),t)-b*d(argument(f,1),x),y), replace(replace(dsolve(-f/b,y,x,1), C_1,C_1+C_2*(t-int(1/b,x))),c_1,C_1+C_2*(t-int(1/b,x))) ))); #pdsolve(y(p_,t_)^m_+y(p_,x_)^m_+c_,y_):= if(hasnot(c,t) and hasnot(c,x), replace(ode(y(p,x)^m+c,y),x,(t+x)/2^(1/m)) ); #pdsolve(y(p_,t_)^m_+b_*y(p_,x_)^m_+c_,y_):= if(hasnot(c,t) and hasnot(c,x), replace(ode(y(p,x)^m+c/(1+b),y),x,t+x) ); pdsolve(a_+y(1,t_)+y_*y(1,x_),y_):= if(isconstant(a), y^2/2+y+C_1+a*t+a*x=0 ); pdsolve(a_+y(1,t_)+b_*y_*y(1,x_),y_):= if(isconstant(a,b), y^2/2+y+C_1+a*t+a*x/b=0 ); pdsolve(a_+y(1,t_)+b_*log(y_)*y(1,x_),y_):= if(isconstant(a,b), log(y)*y=C_1-a*t-a*x/b ); pdsolve(y(p_,t_)^m_+y(q_,x_)^n_+c_*y_,y_):= if(hasnot(c,t) and hasnot(c,x) and hasnot(y), ode(y(p,t)^m+c*y,y)+ode(y(q,x)^n+c*y,y) ); pdsolve(y(p_,t_)^m_+b_*y(q_,x_)^n_+c_*y_,y_):= if(hasnot(b,c,t) and hasnot(b,c,x) and hasnot(b,y), ode(y(p,t)^m+c*y,y)+ode(b*y(q,x)^n+c*y,y) ); pdsolve(y(p_,t_)+y(q_,x_)^n_+c_*y_,y_):= if(isconstant(c), ode((-c*y)^(1/n),y,x,q)+ode(-c*y,y,t,p) ); pdsolve(y(p_,t_)+y(q_,x_)^n_+y_,y_):= ode((-y)^(1/n),y,x,q)+ode(-y,y,t,p); pdsolve(y(p_,t_)+y(q_,x_)+c_*y_,y_):= if(isconstant(c), ode(-c*y,y,x,q)+ode(-c*y,y,t,p) ); pdsolve(y(p_,t_)+y(q_,x_)+y_,y_):= ode(-y,y,x,q)+ode(-y,y,t,p); #pdsolve(y_+y(p_,t_)+y(q_,x_),y_):=if(p<=1 and q<=1,C_2+t^p/p!-x^q/q!,1)*gsolution(-1,y,t,p)*if(p<=1 and q>1,gsolution(0,y,x,q),1) +if(q<=1 and p<=1,C_2+t^p/p!-x^q/q!,1)*gsolution(-1,y,x,q)*if(q<=1 and p>1,gsolution(0,y,t,p),1); pdsolve(y_+y(p_,t_)+c_*y(q_,x_),y_):=if(p<=1 and q<=1,if(hasnot(c,x),C_2+int(c,t,p)-x^q/q!,C_2+t^p/p!-int(1/c,x,q)),1)*gsolution(-1,y,t,p) +if(p<=1 and q<=1,C_2+t^p/p!-int(1/c,x,q),1)*if(hasnot(c,t),gsolution(-1/c,y,x,q),0); pdsolve(y_+b_*y(p_,t_)+c_*y(q_,x_),y_):=if(hasnot(b,x),gsolution(-1/b,y,t,p),0)*if(p<=1 and q<=1 and hasnot(c,t),(C_2+int(1/b,t,p)-int(1/c,x,q)),1)*if(p<=1 and q>1,gsolution(0,y,x,q),1) +if(hasnot(c,t),gsolution(-1/c,y,x,q),0)*if(p<=1 and q<=1 and hasnot(b,x),(C_2+int(1/b,t,p)-int(1/c,x,q)),1); pdsolve(a_*y_+b_*y(p_,t_)+c_*y(q_,x_),y_):=if(hasnot(a,b,x),gsolution(-a/b,y,t,p),0)*if(p<=1 and q<=1 and hasnot(c,t),(C_2+int(1/b,t,p)-int(1/c,x,q)),1)*if(p<=1 and q>1,gsolution(0,y,x,q),1) +if(hasnot(a,c,t),gsolution(-a/c,y,x,q),0)*if(p<=1 and q<=1 and hasnot(b,x),(C_2+int(1/b,t,p)-int(1/c,x,q)),1); #pdsolve(a_*y_+y(p_,t_)+c_*y(q_,x_),y_):=if(hasnot(a,x),gsolution(-a,y,t,p),0)*if(p<=1 and q<=1,C_2+if(hasnot(c,x),int(c,t,p),t^p/p!)-if(has(c,x),int(1/c,x,q),x^q/q!/q),1) +if(hasnot(a,t),gsolution(-a/c,y,x,q),0)*if(p<=1 and q<=1,C_2+if(hasnot(c,x),int(c,t,p),t^p/p!)-if(has(c,x),int(1/c,x,q),x^q/q!/q),1); #pdsolve(a_*y_+y(p_,t_)+c_*y(q_,x_),y_):=if(p<=1 and q<=1 and isconstant(a),if(hasnot(c,x),C_2+int(c,t,p)-x^q/q!,C_2+t^p/p!-int(1/c,x,q)),1)*if(hasnot(a,x),gsolution(-a,y,t,p),0)*if(p<=1 and q>1,gsolution(0,y,x,q),1) +if(p==1 and q==1 and isconstant(c),C_1*exp(-int(replace(replace(a,t,z),x,x-c*t+c*z),z,0,t))*F(t-x/c),0) +if(p<=1 and q<=1 and isconstant(c),if(hasnot(c,x),C_2+int(c,t,p)-x^q/q!,C_2+t^p/p!-int(1/c,x,q),1)*if(hasnot(a/c,t),gsolution(-a/c,y,x,q),0),0); pdsolve(a_*y_+y(p_,t_)+c_*y(q_,x_),y_):=if(p<=1 and q<=1 and isconstant(a),if(hasnot(c,x),C_2+int(c,t,p)-x^q/q!,C_2+t^p/p!-int(1/c,x,q)),1)*if(hasnot(a,x),gsolution(-a,y,t,p),0)*if(p<=1 and q>1,gsolution(0,y,x,q),1) +if(p==1 and q==1 and isconstant(c),C_1*exp(-int(replace(replace(a,t,z),x,x-c*t+c*z),z,0,t))*(t-x/c),0) +if(p<=1 and q<=1 and isconstant(c),if(hasnot(c,x),C_2+int(c,t,p)-x^q/q!,C_2+t^p/p!-int(1/c,x,q))*if(hasnot(a/c,t),gsolution(-a/c,y,x,q),0),0); #pdsolve(a_*y_+y(p_,t_)+y(q_,x_),y_):=if(p<=1 and q<=1,C_2+t^p/p!-x^q/q!,1)*if(hasnot(a,x),gsolution(-a,y,t,p),0) +if(p<=1 and q<=1,C_2+t^p/p!-x^q/q!,1)*if(hasnot(a,t),gsolution(-a,y,x,q),0) +if(p<=1 and q<=1,(t^p/p!-x^q/q!),1)*if(has(a,t) and has(a,x),gsolution(-replace(a,x,0),y,t,p)*gsolution(-replace(a,t,0),y,x,q),0); #pdsolve(a_*y_+y(p_,t_)+y(q_,x_),y_):= C_1*exp(psolution(1,q,0,a,y,t,x,p))*(x^q/q!-t^p/p!); pdsolve(c_*y_^3+y(1,t_)+y(2,x_),y_):= sqrt(-2/c)*(2C_1*x+C_2)/(C_1*x^2+C_2*x-6C_1*t+C_3); pdsolve(c_*y_^3+y(1,t_)-y(2,x_),y_):= sqrt(2/c)*(2C_1*x+C_2)/(C_1*x^2+C_2*x+6C_1*t+C_3); pdsolve(y_+y(1,t_)+y_*y(1,x_),y_):= log(y)+y=C_1-t-x; pdsolve(a_*y_+y(1,t_)+y_*y(1,x_),y_):= if(isconstant(a), log(y)/a+y=C_1-t and C_1-a*x); #pdsolve(a_*y_+y(1,t_)+c_*y(1,x_),y_):= if(hasnot(a,y), C_1*exp(psolution(c,1,0,a,y,t,x,1))*(if(has(c,t),int(c,t),t)-if(hasnot(c,t),int(1/c,x),x)) ); #pdsolve(a_*y_+y(1,t_)+y(1,x_),y_):= if(hasnot(a,y), C_1*exp(-int(replace(replace(a,t,z),x,x-t+z),z,0,t))*(t-x) ); pdsolve(a_*log(y_)*y_+y(2,t_)+y(2,x_),y_):= exp(-a*(t+C_1)^2/4-a*(x+c_1)^2/4+1); pdsolve(a_*log(y_)*y_+y(2,t_)+b_*y(2,x_),y_):= if(isconstant(b), exp(-a*(t+C_1)^2/4-a*(x+c_1)^2/4/b+1) ); pdsolve(a_*log(y_)*y_+y(1,t_)+y(2,x_),y_):= exp(C_1*exp(-a*t)-(x+c_1)^2/4/a+1/2); pdsolve(a_*log(y_)*y_+y(1,t_)+b_*y(2,x_),y_):= if(isconstant(b), exp(C_1*exp(-a*t)-(x+c_1)^2/4/a/b+1/2) ); pdsolve(log(y_)*y_+y(1,t_)+y(2,x_),y_):= exp(C_1*exp(-t)+1/2-(x+c_1)^2/4); pdsolve(log(y_)*y_+y(1,t_)+b_*y(2,x_),y_):= if(isconstant(b), exp(C_1*exp(-t)+1/2-(x+c_1)^2/4/b) ); pdsolve(y(p_,t_)+b_*y(p1_,x_)+c_*y(p2_,u_),y_):= if(isconstant(b,c), C_1+C_2*(2*t^p/p!-x^p1/p1!/b-u^p2/p2!/c) ); #pdsolve(y(p_,t_)+b_*y(2,x_)+c_*y(2,u_),y_):= if(isconstant(b,c), C_1+C_3*(2t^p/p!-x*x/2/b-u*u/2/c)+C_2*E(p,(b+c)*t^p)*sin(x)*sin(u) ); pdsolve(y(p_,t_)^m_+b_*y(p_,x_)^m_+y(p_,x_),y_):= if(hasnot(b,t) and hasnot(b,x), replace(ode(y(p,x)^m+y(p,x)/(1+b),y),x,t+x) ); pdsolve(y(p_,t_)^m_+b_*y(p_,x_)^m_+c_*y(p_,x_),y_):= if(hasnot(b,c,t) and hasnot(b,c,x), replace(ode(y(p,x)^m+c*y(p,x)/(1+b),y),x,t+x) ); pdsolve(y(p_,t_)^m_+y(p_,x_)^m_+c_*y(p_,x_),y_):= if(hasnot(c,t) and hasnot(c,x), replace(ode(y(p,x)^m+c*y(p,x)/2,y),x,t+x) ); pdsolve(y(p_,t_)+a_*y(1,x_)^m_+y(q_,x_),y_):= ode(-a*y(1,x)^m,y,x,q)+if(hasnot(a,t),gsolution(0,y,t,p),0) and if(isconstant(a),C_1*x-a*C_1^2*t^p/p!+gsolution(0,y,t,p)); pdsolve(y(p_,t_)+y(1,x_)^m_+b_*y(q_,x_),y_):= if(hasnot(b,y),ode(b*y(q,x)+y(1,x)^m-1,y)-t^p/p!+gsolution(0,y,t,p),ode(-1/b*y(1,x)^m,y,x,q)*gsolution(0,y,t,p)) and C_1*x-C_1^2*t^p/p!+gsolution(0,y,t,p); pdsolve(y(p_,t_)+a_*y(1,x_)^m_+b_*y(q_,x_),y_):= if(hasnot(a,x), C_1*x-C_1^m*int(a,t,p)+gsolution(0,y,t,p), if(hasnot(a,b,t), gsolution(0,y,t,p)+(int((-b/a)^(1/(m-1)),x,q)) )); pdsolve(y(p_,t_)+y(q_,x_)^m_+y(q_,x_),y_):= C_1*(x-t^p/p!)-C_1^m*t^p/p!+gsolution(0,y,t,p); pdsolve(y(p_,t_)+y(q_,x_)^m_+b_*y(q_,x_),y_):= if(has(b,y), ode((-b)^(1/(m-1)),y,x,q), if(hasnot(b,x), C_1*(x-int(b,t,p))-C_1^m*t^p/p!+gsolution(0,y,t,p), if(hasnot(b,t), int((-b)^(1/(m-1)),x,q)+gsolution(0,y,t,p) ))); pdsolve(y(p_,t_)+a_*y(q_,x_)^m_+y(q_,x_),y_):= if(has(a,y), ode((-1/a)^(1/(m-1)),y,x,q), if(hasnot(a,x), C_1*(x-t^p/p!)-C_1^m*int(a,t,p)+gsolution(0,y,t,p), if(hasnot(a,t), int((-1/a)^(1/(m-1)),x,q)+gsolution(0,y,t,p) ))); pdsolve(y(p_,t_)+a_*y(q_,x_)^m_+b_*y(q_,x_),y_):= if(has(b,y), ode((-b/a)^(1/(m-1)),y,x,q), if(hasnot(a,b,x), C_1*(x-int(b,t,p))-C_1^m*int(a,t,p)+gsolution(0,y,t,p), if(hasnot(a,b,t), int((-b/a)^(1/(m-1)),x,q)+gsolution(0,y,t,p) ))); pdsolve(y(p_,t_)+(a_+c_)*y(1,x_)^m_+b_*y(1,x_),y_):= if(hasnot(a,b,c,t) and hasnot(a,b,c,x), int((a+c)/(a+b),y)+a*t^p/p!+x=C_1 ); #pdsolve(y(p_,t_)+a_*y(1,x_)^2+b_*y(2,x_),y_):= if(isconstant(a), if(hasnot(b,y),ode(b*y(2,x)+a*y(1,x)^2-1,y)-t^p/p!+gsolution(0,y,t,p), ode(b*y(2,x)+a*y(1,x)^2,y)*gsolution(0,y,t,p) ) and C_1*x-a*C_1^2*t^p/p!+gsolution(0,y,t,p), if(d(b,y)==a, if(p==1, int(b/y,y)=C_1-t+x, if(p==2, b=-(C_1+x)^2/(C_2+t)^2 )))); pdsolve(y(p_,t_)+a_*y(1,x_)^m_+y_*y(q_,x_),y_):= if(isconstant(a),x-a*t^p/p!+gsolution(0,y,t,p) ); pdsolve(y(p_,t_)+a_*y(1,x_)^m_+b_*y_*y(q_,x_),y_):= if(isconstant(a),x-a*t^p/p!+gsolution(0,y,t,p) ); pdsolve(y(p_,t_)+exp(y_)*y(1,x_)^2+exp(y_)*y(2,x_),y_):= -p*log(C_1+t)+log((-1)^(p+1)*p!*x^2/2); pdsolve(y(p_,t_)+b_*exp(y_)*y(1,x_)^2+b_*exp(y_)*y(2,x_),y_):= -p*log(C_1+t)+log((-1)^(p+1)*p!*x^2/2/b); pdsolve(y(p_,t_)+y(m_,x_)+y(q_,x_),y_):= gsolution(-1,m,0,y,x,q)+gsolution(0,y,t,p)+C_3*(t^p/p!-x^m/m!); pdsolve1(y(p_,t_)+y(m_,x_)+b_*y(q_,x_),y_):= if(hasnot(b,y), gsolution(-1/b,m,0,y,x,q)+gsolution(0,y,t,p)+C_3*(t^p/p!-x^m/m!) ); pdsolve1(y(p_,t_)+a_*y(m_,x_)+y(q_,x_),y_):= if(hasnot(a,y), gsolution(-a,m,0,y,x,q)+gsolution(0,y,t,p)+C_3*(t^p/p!-if(isconstant(a),x^m/m!/a,psolution(-a,m,0,1,y,x,q))) ); pdsolve1(y(p_,t_)+a_*y(m_,x_)+b_*y(q_,x_),y_):= if(m==q, pdsolve(y(p,t)+(a+b)*y(m,x),y), if(hasnot(a,b,y), gsolution(-a/b,m,0,y,x,q)+gsolution(0,y,t,p)+C_3*(t^p/p!-if(isconstant(a),x^m/m!/a,psolution(-a/b,m,0,1/b,y,x,q))) )); pdsolve(y(p_,t_)+y(q_,x_)+b_*y(1,x_),y_):= if(isconstant(b), C_1*(x-b*t^p/p!)+C_3*exp(-int(b,x))+C_2, if(hasnot(b,x), pde(y(p,t)+b*y(1,x),y) )); pdsolve(y(p_,t_)+a_*y(q_,x_)+b_*y(1,x_),y_):= if(isconstant(a,b), C_1*(x-b*t^p/p!)+C_3*exp(-int(b/a,x))+C_2, if(hasnot(b,x), pde(y(p,t)+b*y(1,x),y) )); pdsolve(y(p_,t_)+a_*y(2,x_)+y(1,x_),y_):= if(has(a,y) and hasnot(a,t) and hasnot(a,x), C_2+C_1*(t^p/p!-x) and if(p==1, int(1/int(1/a,y),y)=C_1-2t+x )); pdsolve(y(p_,t_)+a_*y(2,x_)+b_/x_*y(1,x_),y_):= if(isconstant(a,b), gsolution(-b/a/x,1,0,y,x,2)+gsolution(0,y,t,p)+C_3*(t^p/p! -if(a+b<>0,1/(a+b)*x^2/2,1/a*((-1/4)*x^2+1/2*log(x)*x^2))) +if(p==1 and a==b,C_4*Ei(x^2/(2*(a+b)*t)),0) ); pdsolve(y(p_,t_)+y(q_,x_)+b_*y_*y(m_,x_),y_):= if(isconstant(b),-fallingfactorial(-p,p)*(C_1+t)^(-p)*(C_2+x)^m/(m-1)!/b, if(hasnot(b,x), (-1)^(1+p)*(C_2+x)^m/(m-1)!/(C_1+int(b,t,p)) )); pdsolve(y(p_,t_)+a_*y(q_,x_)+y_*y(m_,x_),y_):= if(m==q,-a,0)-fallingfactorial(-p,p)*(C_1+t)^(-p)*(C_2+x)^m/m!; pdsolve(y(p_,t_)+y(q_,x_)+y_*y(m_,x_),y_):= if(m==q,-1,0)-fallingfactorial(-p,p)*(C_1+t)^(-p)*(C_2+x)^m/m!; pdsolve(y(p_,t_)+a_*y(q_,x_)+b_*y_*y(m_,x_),y_):= if(isconstant(b),if(m==q,-a/b,0)-fallingfactorial(-p,p)*(C_1+t)^(-p)*(C_2+x)^m/(m-1)!/b, if(hasnot(b,x), (-1)^(1+p)*(C_2+x)^m/(m-1)!/(C_1+int(b,t,p)) )); #pdsolve(y(1,t_)+a_*x_*y(2,x_)+a_*y(1,x_),y_):= if(isconstant(a), C_1+C_2*log(x)+C_3*(t-1/a*x)+C_4*Ei(x/(a*t)) ); pdsolve(y(1,t_)+y_*y(2,x_)+y(1,x_),y_):= inverseli(C_1-2t+x) and C_2+C_1*(t-x); pdsolve(y(1,t_)+a_*y_*y(2,x_)+y(1,x_),y_):= if(isconstant(a), inverseli(C_1-(1+a)*t+x) and C_2+C_1*(t-x)); pdsolve(y(1,t_)+a_*y_*y(2,x_)+b_*y(1,x_),y_):= if(isconstant(a,b), inverseli(C_1-(a+b)*t+x) and C_2+C_1*(t-x/b)); #pdsolve(y(1,t_)+a_*y(2,x_)+y_*y(1,x_),y_):= 2a/(C_1+C_2*t+x)-C_2; #pdsolve(y(1,t_)+a_*y(2,x_)+b_*y_*y(1,x_),y_):= 2a/b/(C_1+C_2*b*t+x)-C_2; #pdsolve(y(1,t_)+a_*x_*y(1,x_)+b_*y(1,x_),y_):= F(exp(-int(a,t))*x-int(exp(-int(a,t))*b,t)); #pdsolve(y(1,t_)+a_*x_*y(1,x_)+b_*x_^n_*y(1,x_),y_):= F(exp((n-1)*int(a,t))*x^(1-n)+(n-1)*int(exp((n-1)*int(a,t))*b,t)); #pdsolve(y(1,t_)+y(3,x_)+c_*y(1,x_),y_):= if(hasnot(c,t) and hasnot(c,x), solve(c=1/(C_1+t)^2+(C_2+x)/(C_1+t),y) ); #pdsolve(y(1,t_)+b_*y(3,x_)+c_*y(1,x_),y_):= if(hasnot(b,c,t) and hasnot(b,c,x), solve(c=b/(C_1+t)^2+(C_2+x)/(C_1+t),y) ); pdsolve(y(1,t_)+b_*y(3,x_)+c_*acosh(y_)*y(1,x_),y_):= if(isconstant(b,c),cosh(b/c/(C_1+c*t)^2+(C_2+x)/(C_1+c*t)) ); pdsolve(y(1,t_)+b_*y(3,x_)+c_*log(y_)*y(1,x_),y_):= if(isconstant(b,c),exp(b/c/(C_1+c*t)^2+(C_2+x)/(C_1+c*t)) ); pdsolve(y(1,t_)+b_*y(3,x_)+log(y_)*y(1,x_),y_):= exp(b/(C_1+t)^2+(C_2+x)/(C_1+t)); #pdsolve(y(1,t_)+y(3,x_)+c_*y_^2*y(1,x_),y_):= C_1+C_2^2/(sqrt(4C_1^2+C_2^2)*cosh(C_2*x-(c*C_1^2*C_2+C_2^3)*t+C_3)+2C_1); pdsolve(y(1,t_)+y(3,x_)+c_*y_^2*y(1,x_),y_):= C_1-4C_1/(4C_1^2*(x-c*C_1^2*t)^2+1); #4; pdsolve(y(p_,t_)^k_+y(q_,x_)^n_+b_*y(m_,x_)+d_,y_):= if(hasnot(d,b,y),if(hasnot(d,x), ode(y(q,x)^n+b*y(m,x),y)+int((-d)^(1/k),t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+b*y(m,x)+d,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)^k_+a_*y(q_,x_)^n_+y(m_,x_)+d_,y_):= if(hasnot(a,d,y),if(hasnot(d,x), ode(y(q,x)^n+1/a*y(m,x),y)+int((-d)^(1/k),t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+1/a*y(m,x)+1/a,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)^k_+a_*y(q_,x_)^n_+b_*y(m_,x_)+d_,y_):= if(hasnot(a,d,b,y),if(hasnot(d,x), ode(y(q,x)^n+b/a*y(m,x),y)+int((-d)^(1/k),t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+b/a*y(m,x)+d/a,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)+y(q_,x_)^n_+y(m_,x_)+d_,y_):= if(hasnot(d,y),if(hasnot(d,x), ode(y(q,x)^n+y(m,x),y)-int(d,t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+y(m,x)+d,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)+y(q_,x_)^n_+b_*y(m_,x_)+d_,y_):= if(hasnot(d,b,y),if(hasnot(d,x), ode(y(q,x)^n+b*y(m,x),y)-int(d,t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+b*y(m,x)+d,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)+a_*y(q_,x_)^n_+y(m_,x_)+d_,y_):= if(hasnot(a,d,y),if(hasnot(d,x), ode(y(q,x)^n+1/a*y(m,x),y)-int(d,t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+1/a*y(m,x)+1/a,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)+a_*y(q_,x_)^n_+b_*y(m_,x_)+d_,y_):= if(hasnot(a,d,b,y),if(hasnot(d,x), ode(y(q,x)^n+b/a*y(m,x),y)-int(d,t,p)+gsolution(0,y,t,p),ode(y(q,x)^n+b/a*y(m,x)+d/a,y)+gsolution(0,y,t,p) )); pdsolve(y(p_,t_)+y(q_,x_)+y_+d_,y_):= if(hasnot(d,y), if(p<=1 and q<=1,(C_2+t^p/p!-x^q/q!),1)*gsolution(-1,y,t,p) +if(q<=1,(C_2+t^p/p!-x^q/q!),1)*gsolution(-1,y,x,q) +psolution(1,q,1,d,y,t,x,p) ); pdsolve(y(p_,t_)+y(q_,x_)+b_*y_+d_,y_):= if(hasnot(d,y), if(p<=1 and q<=1,C_2-t^p/p!+x^q/q!,1)*if(hasnot(b,x),gsolution(-b,y,t,p),0) +if(q<=1,C_2+t^p/p!-x^q/q!,1)*if(hasnot(b,t),gsolution(-b,y,x,q),0) +psolution(1,q,b,d,y,t,x,p) ); pdsolve(y(p_,t_)+a_*y(q_,x_)+y_+d_,y_):= if(hasnot(a,d,y), if(p<=1 and q<=1,C_2+if(hasnot(a,x),int(a,t,p),t^p/p!)-if(hasnot(a,t),int(1/a,x,q),x^q/q!),1)*gsolution(-1,y,t,p) +if(q<=1,C_2+t^p/p!-int(1/a,x,q),1)*if(hasnot(a,t),gsolution(-1/a,y,x,q),0) +psolution(a,q,1,d,y,t,x,p) ); pdsolve(y(p_,t_)+a_*y(q_,x_)+b_*y_+d_,y_):=if(hasnot(a,b,d,y), if(p<=1 and q<=1,C_2-if(hasnot(a,x),int(a,t,p),t^p/p!)+if(hasnot(a,t),int(1/a,x,q),x^q/q!),1)*if(hasnot(b,x),gsolution(-b,y,t,p),0) +if(q<=1,C_2+t^p/p!-int(1/a,x,q),1)*if(hasnot(a,b,t),gsolution(-b/a,y,x,q),0) +psolution(a,q,b,d,y,t,x,p) ); pdsolve(y(1,t_)+y(2,x_)+b_*exp(y_)+d_,y_):= if(isconstant(b,d), -2log(sqrt(-b/d)+C_1*exp(sqrt(d/2)*x+d*t/2)) ); pdsolve(y(1,t_)+a_*y(2,x_)+b_*exp(y_)+d_,y_):= if(isconstant(a,b,d), -2log(sqrt(-b/d)+C_1*exp(sqrt(-d/a/2)*x+d*t/2)) ); pdsolve(a_*y_+b_*y_+y(p_,t_)+y(q_,x_),y_):=if(hasnot(a,x),gsolution(-a,y,t,p),1)*if(hasnot(b,t),gsolution(-b,y,x,q),gsolution(-b,y,t,p))*if(p==q and p==1,C_2+t-x,1); pdsolve(a_*y_+b_*y_+y(p_,t_)+d_*y(q_,x_),y_):=if(hasnot(a,x),gsolution(-a,y,t,p),1)*if(hasnot(b,t),gsolution(-b/d,y,x,q),gsolution(-b,y,t,p))*if(p==q and p==1,C_2+t-int(1/d,x,q),1); #pdsolve(y(q_,t_)+y(q_,x_)+c_*y_+d_*y_^n_,y_):= if(hasnot(c,t) and hasnot(c,x), block(f:=dsolve(-c*y/2-d*y^n/2,y,x,q),if(has(f,x^q), replace(f,x^q,t^q+x^q),replace(f,x,c_1+C_2*(t+x)))) ); pdsolve(y(1,t_)+y(2,x_)+y_+d_*y_^n_,y_):= if(hasnot(d,t) and hasnot(d,x), (sqrt(-d)+C_1*exp(-(1-n)*(3+n)/2/(1+n)*t+sqrt((1-n)^2/2/(1+n))*x))^(2/(1-n)) ); pdsolve(y(1,t_)+b_*y(2,x_)+y_+d_*y_^n_,y_):= if(hasnot(b,d,t) and hasnot(b,d,x), (sqrt(-d)+C_1*exp(-(1-n)*(3+n)/2/(1+n)*t+sqrt(1/b*(1-n)^2/2/(1+n))*x))^(2/(1-n)) ); pdsolve(y(1,t_)+y(2,x_)+c_*y_+d_*y_^n_,y_):= if(hasnot(b,c,d,t) and hasnot(b,c,d,x), (sqrt(-d/c)+C_1*exp(-c*(1-n)*(3+n)/2/(1+n)*t+sqrt(-c*(1-n)^2/2/(1+n))*x))^(2/(1-n)) ); pdsolve(y(1,t_)+b_*y(2,x_)+c_*y_+d_*y_^n_,y_):= if(hasnot(b,c,d,t) and hasnot(b,c,d,x), (sqrt(-d/c)+C_1*exp(-c*(1-n)*(3+n)/2/(1+n)*t+sqrt(c/b*(1-n)^2/2/(1+n))*x))^(2/(1-n)) ); pdsolve(y(1,t_)+y(2,x_)+c_*y_^2+d_*y_^3,y_):= if(c+d==0, 1/(exp(sqrt(c/2)*x+d/2t)+C_1) ); pdsolve(y(1,t_)+b_*y(2,x_)+c_*y_^2+d_*y_^3,y_):= if(c+d==0, 1/(exp(sqrt(d/b/2)*x+d/2t)+C_1) ); pdsolve(y(2,t_)+b_*y(2,x_)+c_*y_^m_+d_*y_^n_,y_):= if(2m-1==n, (-1/4*c*(1-m)^2*((t+C_1)^2-(x+C_2)^2)-d/c/m)^(1/(1-m)) ); pdsolve(y(2,t_)+b_*y(2,x_)+c_*y_+d_*y_^n_,y_):= (2d/c/(1+n)*sinh(1/2*sqrt(-c)*(1-n)*(sinh(C_1)*x+cosh(C_1)*t)+C_2)^2)^(1/(1-n)); pdsolve(y(p_,t_)+b_*y(p1_,x_)+c_*y(p2_,u_)+f_,y_):= if(isconstant(b,c), psolution(b,p1,c,p2,0,f,y,t,x,u,p)+C_1+C_2*(2*t^p/p!-x^p1/p1!/b-u^p2/p2!/c) ); pdsolve(y(p_,t_)+y(n_,x_)+y(m_,x_)+d_,y_):= if(hasnot(d,y), C_3*(1/p!*t^p-1/m!*x^m)+gsolution(-1,m,0,y,x,n) +psolution(1,n,1,m,0,d,y,t,x,p) ); pdsolve(y(p_,t_)+a_*y(n_,x_)+y(m_,x_)+d_,y_):= if(hasnot(a,d,y), C_3*(1/p!*t^p-1/m!*x^m)+gsolution(-1/a,m,0,y,x,n) +psolution(a,n,1,m,0,d,y,t,x,p) ); pdsolve(y(p_,t_)+y(n_,x_)+b_*y(m_,x_)+d_,y_):= if(hasnot(b,d,y), pdsolve(y(p,t)+y(n,x)+b*y(m,x),y) +psolution(1,n,b,m,0,d,y,t,x,p) ); pdsolve(y(p_,t_)+a_*y(n_,x_)+b_*y(m_,x_)+d_,y_):= if(hasnot(a,b,d,y), pdsolve(y(p,t)+a*y(n,x)+b*y(m,x),y) +psolution(a,n,b,m,0,d,y,t,x,p) ); #pdsolve(y(p_,t_)^2+y(q_,x_)^2+a_+c_/y_^2,y_):= if(hasnot(a,c,t) and hasnot(a,c,x), if(a==-1,i*sqrt(-c-(t+C_1)^2-(x+C_2)^2), if(a==1,sqrt(-c-(t+C_1)^2-(x+C_2)^2), sqrt(-2 a^2* C_1 *t* x - 2 a^2* C_2 *t - a^2 *C_1^2 *x^2 - 2 a^2* C_1 *C_2* x - a^2 *C_2^2- a^2 *t^2 - c *C_1^2 - c)/sqrt(a *C_1^2 + a) ))); pdsolve(y(p_,t_)^2+y(q_,x_)^2+a_+c_/y_^2,y_):= if(hasnot(a,c,t) and hasnot(a,c,x), if(a==-1,i*sqrt(-c-(t+C_1)^2-(x+C_2)^2), if(a==1,sqrt(-c-(t+C_1)^2-(x+C_2)^2) ))); pdsolve(y(p_,t_)+a_*y(q_,x_)^n_+b_*y(m_,x_)+c_*y_,y_):= if(isconstant(c), ode(a*y(q,x)^n+b*y(m,x)+c*y,y)+if(has(b,x),ode(-c*y,y,t,p),0) ); pdsolve(y(p_,t_)+a_*y(q_,x_)^n_+b_*y(m_,x_)+y_,y_):= ode(a*y(q,x)^n+b*y(m,x)+y,y)+if(has(b,x),ode(-y,y,t,p),0); pdsolve(y(p_,t_)+y(q_,x_)^n_+b_*y(m_,x_)+y_,y_):= ode(y(q,x)^n+b*y(m,x)+y,y)+if(has(b,x),ode(-y,y,t,p),0); pdsolve(y(p_,t_)+y(q_,x_)^n_+b_*y(m_,x_)+c_*y_,y_):= if(isconstant(c), ode(y(q,x)^n+b*y(m,x)+c*y,y)+if(has(b,x),ode(-c*y,y,t,p),0) ); pdsolve(y(p_,t_)+y(1,x_)^2+b_*y(q_,x_)+y_,y_):= -(c_1+x+exp(-t)/((2)^p-1)/(-1)^p)*exp(-t) and -(c_1-x+exp(-t)/((2)^p-1)/(-1)^p)*exp(-t); pdsolve(y(p_,t_)+y(1,x_)^2+b_*y(q_,x_)+c_*y_,y_):=if(isconstant(c), -(c_1+x+exp(-c*t)/((2)^p-1)/(-c)^p)*exp(-c*t)/a and -(c_1-x+exp(-c*t)/((2)^p-1)/(-c)^p)*exp(-c*t)/a ); pdsolve(y(p_,t_)+a_*y(1,x_)^2+b_*y(q_,x_)+y_,y_):=if(isconstant(a), -(c_1+x+exp(-t)/((2)^p-1)/(-1)^p)*exp(-t)/a and -(c_1-x+exp(-t)/((2)^p-1)/(-1)^p)*exp(-t)/a ); pdsolve(y(p_,t_)+a_*y(1,x_)^2+y(q_,x_)+c_*y_,y_):=if(isconstant(c), -(c_1+x+exp(-c*t)/((2)^p-1)/(-c)^p)*exp(-c*t)/a and -(c_1-x+exp(-c*t)/((2)^p-1)/(-c)^p)*exp(-c*t)/a ); pdsolve(y(p_,t_)+a_*y(1,x_)^2+b_*y(q_,x_)+c_*y_,y_):=if(isconstant(c), -(c_1+x+exp(-c*t)/((2)^p-1)/(-c)^p)*exp(-c*t)/a and -(c_1-x+exp(-c*t)/((2)^p-1)/(-c)^p)*exp(-c*t)/a ); pdsolve(y(p_,t_)+b_*y(p1_,x_)+c_*y(p2_,u_)+y_,y_):= gsolution(-1,y,t,p)+gsolution(-1/b,y,x,p1)+gsolution(-1/c,y,u,p2); pdsolve(y(p_,t_)+b_*y(p1_,x_)+c_*y(p2_,u_)+g_*y_,y_):= gsolution(-g,y,t,p)+gsolution(-g/b,y,x,p1)+gsolution(-g/c,y,u,p2); #pdsolve(y(p_,t_)+b_*y(2,x_)+c_*y(2,u_)+g_*y_,y_):= gsolution(-g,y,t,p)+gsolution(-g/b,y,x,2)+gsolution(-g/c,y,u,2); pdsolve(y(p_,t_)+y(n_,x_)+y(m_,x_)+c_*y_,y_):= if(hasnot(c,x),gsolution(-c,y,t,p)*(C_2+exp((-1)^(1/(m-n))*x)),0) +if(hasnot(c,t),gsolution(-1,m,-c,y,x,n),0); pdsolve(y(p_,t_)+y(n_,x_)+b_*y(m_,x_)+y_,y_):= gsolution(-1,y,t,p)*(C_2+if(isconstant(b),exp((-b)^(1/(n-m))*x),0)) +if(p<=1 and has(b,t),C_3*exp(-int(b,t,p)*(-1)^(1/n)+(-1)^(1/n)*x),0) +if(hasnot(b,t),gsolution(-b,m,-1,y,x,n),0); pdsolve(y(p_,t_)+y(n_,x_)+b_*y(m_,x_)+c_*y_,y_):= if(hasnot(c,x),gsolution(-c,y,t,p)*(C_2+if(isconstant(b),exp((-b)^(1/(n-m))*x),0)),0) +if(p<=1 and has(b,t),C_3*exp(int(b,t,p)*(c)^m-int(c,x)),0) +if(hasnot(b,c,t),gsolution(-b,m,-c,y,x,n),0); pdsolve(y(p_,t_)+a_*y(n_,x_)+y(m_,x_)+c_*y_,y_):= if(hasnot(c,x),gsolution(-c,y,t,p)*(C_2+if(isconstant(a),exp((-a)^(1/(m-n))*x),0)),0) +if(p<=1 and has(a,t),C_3*exp((-1)^(m+n)*int(a,t,p)*(c)^n-int(c,x)),0) +if(hasnot(a,c,t),gsolution(-1/a,m,-c/a,y,x,n),0); pdsolve(y(p_,t_)+a_*y(n_,x_)+b_*y(m_,x_)+y_,y_):= gsolution(-1,y,t,p)*(C_2+if(isconstant(a,b),exp((-b/a)^(1/(n-m))*x),0)) +if(p<=1 and has(a,t),C_3*exp((-1)^(m+n)*int(a,t,p)*(1/b)^n-int(1/b,x)),0) +if(p<=1 and has(b,t),C_3*exp(int(b,t,p)*(1/a)^m-int(1/a,x)),0) +if(hasnot(a,b,t),gsolution(-b/a,m,-1/a,y,x,n),0); pdsolve(y(p_,t_)+a_*y(n_,x_)+b_*y(m_,x_)+c_*y_,y_):= if(hasnot(c,x),gsolution(-c,y,t,p)*(C_2+if(isconstant(a,b),exp((-b/a)^(1/(n-m))*x),0)),0) +if(p<=1 and has(a,t),C_3*exp((-1)^(m+n)*int(a,t,p)*(c/b)^n-int(c/b,x)),0) +if(p<=1 and has(b,t),C_3*exp(int(b,t,p)*(c/a)^m-int(c/a,x)),0) +if(hasnot(a,b,c,t),gsolution(-b/a,m,-c/a,y,x,n),0); pdsolve(y_+y(1,t_)+y(2,t_)+y(2,x_),y_):=C_1*cos(1/2*sqrt(3)*x)*exp((-1/2)*t)+C_2*exp((-1/2)*t)*sin(1/2*sqrt(3)*x); pdsolve(c_*y_+y(1,t_)+y(2,t_)+y(2,x_),y_):=if(isconstant(c),C_1*cos(1/2*sqrt(4c-1)*x)*exp((-1/2)*t)+C_2*exp((-1/2)*t)*sin(1/2*sqrt(4c-1)*x)); pdsolve(c_*y_+b_*y(1,t_)+y(2,t_)+y(2,x_),y_):=if(isconstant(b,c),C_1*cos(1/2*sqrt(4c-b^2)*x)*exp((-1/2)*b*t)+C_2*exp((-1/2)*b*t)*sin(1/2*sqrt(4c-b^2)*x)); pdsolve(c_*y_+b_*y(1,t_)+y(2,t_)+a_*y(2,x_),y_):=if(isconstant(a,b,c),C_1*cos(1/2*sqrt(4c/a-b^2/a)*x)*exp((-1/2)*b*t)+C_2*exp((-1/2)*b*t)*sin(1/2*sqrt(4c/a-b^2/a)*x)); pdsolve(y(p_,t_)+b_*y(q_,x_)+c_*y(q_,u_)+d_*y(q_,v_),y_):= C_1+if(isconstant(b,c,d), C_3*(3/p!*t^p-u^q/c/q!-x^q/b/q!-v^q/d/q!+x+u+v),0) +if(q==2,C_2*exp((b+c+d)^(1/p)*t)*sin(x)*sin(u)*sin(v),0 ); pdsolve(y(p_,t_)+b_*y(1,x_)+c_*y(1,u_)+d_*y(1,v_),y_):= if(isconstant(b,c,d), C_1+C_2*(3*t^p/p!-u/c-x/b-v/d) ); pdsolve(a_*y(1,t_)^2+y_*y(2,t_)+a_*y(1,x_)^2+y_*y(2,x_),y_):= if(isconstant(a), replace(ode(a*y(1,x)^2+y*y(2,x),y),x,t+x) ); pdsolve(a_*y(1,t_)^2+b_*y_*y(2,t_)+c_*y(1,x_)^2+d_*y_*y(2,x_),y_):= if(a*d==b*c, replace(ode(a*y(1,x)^2+b*y*y(2,x),y),x,t+x) ); #5; #pdsolve(y(p_,t_)+a_*y(q_,x_)^n_+b_*y(m_,x_)+c_*y_+f_,y_):= if(isconstant(a,c), ode(y(q,x)^n+b/a*y(m,x)+c/a*y,y)+if(has(b,x),ode(-c*y,y,t,p),0)-f/c ); #pdsolve(y(p_,t_)+y(q_,x_)^n_+b_*y(m_,x_)+c_*y_+f_,y_):= if(isconstant(c,f), ode(y(q,x)^n+b*y(m,x)+c*y,y)+if(has(b,x),ode(-c*y,y,t,p),0)-f/c ); pdsolve(y(p_,t_)+y(p_,x_)+a_+b_*y_+y_^2,y_):= if(b*b==4a, pdsolve(y(p,t)+y(p,x)+(y+b/2)^2,y) ); pdsolve(y(p_,t_)+y(p_,x_)+a_+b_*y_+c_*y_^2,y_):= if(b*b==4a*c, pdsolve(y(p,t)+y(p,x)+c*(y+b/2/c)^2,y) ); pdsolve(y(p_,t_)+f_*y(p_,x_)+a_+b_*y_+y_^2,y_):= if(b*b==4a, pdsolve(y(p,t)+f*y(p,x)+(y+b/2)^2,y) ); pdsolve(y(p_,t_)+f_*y(p_,x_)+a_+b_*y_+c_*y_^2,y_):= if(b*b==4a*c, pdsolve(y(p,t)+f*y(p,x)+c*(y+b/2/c)^2,y) ); pdsolve(y(p_,t_)+b_*y(p1_,x_)+c_*y(p2_,u_)+g_*y_+f_,y_):= psolution(b,p1,c,p2,g,f,y,t,x,u,p) +gsolution(-g,y,t,p)+gsolution(-g/b,y,x,p1)+gsolution(-g/c,y,u,p2); pdsolve(y(p_,t_)+y(n_,x_)+y(m_,x_)+y_+d_,y_):= gsolution(-1,y,t,p)*(1+exp((-1)^(1/(n-m))*x))+gsolution(-1,m,-1,y,x,n) +psolution(1,n,1,m,1,d,y,t,x,p); pdsolve(y(p_,t_)+y(n_,x_)+b_*y(m_,x_)+y_+d_,y_):= gsolution(-1,y,t,p)*(1+if(isconstant(b),exp((-b)^(1/(n-m))*x),0))+gsolution(-b,m,-1,y,x,n) +psolution(1,n,b,m,1,d,y,t,x,p); pdsolve(y(p_,t_)+y(n_,x_)+y(m_,x_)+c_*y_+d_,y_):= gsolution(-c,y,t,p)+gsolution(-1,m,-c,y,x,n) +psolution(1,n,1,m,c,d,y,t,x,p); pdsolve(y(p_,t_)+y(n_,x_)+b_*y(m_,x_)+c_*y_+d_,y_):= gsolution(-c,y,t,p)*(1+if(isconstant(b),exp((-b)^(1/(n-m))*x),0))+gsolution(-b,m,-c,y,x,n) +psolution(1,n,b,m,c,d,y,t,x,p); pdsolve(y(p_,t_)+a_*y(n_,x_)+y(m_,x_)+c_*y_+d_,y_):= gsolution(-c,y,t,p)*(1+if(isconstant(a),exp((-a)^(1/(m-n))*x),0))+gsolution(-1/a,m,-c/a,y,x,n) +psolution(a,n,1,m,c,d,y,t,x,p); pdsolve(y(p_,t_)+a_*y(n_,x_)+b_*y(m_,x_)+y_+d_,y_):= gsolution(-1,y,t,p)*(1+if(isconstant(a,b),exp((-b/a)^(1/(n-m))*x),0))+gsolution(-b/a,m,-1/a,y,x,n) +psolution(a,n,b,m,1,d,y,t,x,p); pdsolve(y(p_,t_)+a_*y(n_,x_)+b_*y(m_,x_)+c_*y_+d_,y_):= gsolution(-c,y,t,p)*(1+if(isconstant(a,b),exp((-b/a)^(1/(n-m))*x),0))+gsolution(-b/a,m,-c/a,y,x,n) +psolution(a,n,b,m,c,d,y,t,x,p); pdsolve(y(p_,t_)+b_*y(q_,x_)+c_*y(q_,u_)+d_*y(q_,v_)+f_,y_):= psolution(b,q,c,q,d,q,0,f,y,t,x,u,v,p) +if(isconstant(b,c,d),C_3*(3/p!*t^p-u^q/c/q!-x^q/b/q!-v^q/d/q!),0); pdsolve(y(p_,t_)+b_*y(q_,x_)+c_*y(q_,u_)+d_*y(q_,v_)+y_,y_):= gsolution(-1,y,t,p)+gsolution(-1/b,y,x,q)+gsolution(-1/c,y,u,q)+gsolution(-1/d,y,v,q); pdsolve(y(p_,t_)+b_*y(q_,x_)+c_*y(q_,u_)+d_*y(q_,v_)+g_*y_,y_):= gsolution(-g,y,t,p)+gsolution(-g/b,y,x,q)+gsolution(-g/c,y,u,q)+gsolution(-g/d,y,v,q); pdsolve(y(2,t_)+b_*y(2,x_)+c_*y(1,t_)+d_*y(1,x_)+g_*y_,y_):= gsolution(-c,-g,y,t,2)*(1+exp(-d/b*x))+gsolution(-d/b,-g/b,y,x,2)*(1+exp(-c*t)); pdsolve(y(2,x_)+b_*y(2,t_)+c_*y(1,x_)+d_*y(1,t_)+g_*y_,y_):= gsolution(-c,-g,y,x,2)*(1+exp(-d/b*t))+gsolution(-d/b,-g/b,y,t,2)*(1+exp(-c*x)); pdsolve(y(2,x_)+b_*y(2,t_)+c_*y(1,x_)+d_*y(1,t_)+y_,y_):= gsolution(-c,-1,y,x,2)*(1+exp(-d/b*t))+gsolution(-d/b,-1/b,y,t,2)*(1+exp(-c*x)); pdsolve(y(2,x_)+y(2,t_)+c_*y(1,x_)+d_*y(1,t_)+g_*y_,y_):= gsolution(-c,-g,y,x,2)*(1+exp(-d*t))+gsolution(-d,-g,y,t,2)*(1+exp(-c*x)); pdsolve(y(2,x_)+y(2,t_)+c_*y(1,x_)+d_*y(1,t_)+y_,y_):= gsolution(-c,-1,y,x,2)*(1+exp(-d*t))+gsolution(-d,-1,y,t,2)*(1+exp(-c*x)); pdsolve(y(p_,t_)+b_*y(1,x_)+c_*y(1,u_)+d_*y(1,v_)+g_*y_,y_):= gsolution(-g,y,t,p)+gsolution(-g/b,y,x,1)+gsolution(-g/c,y,u,1)+gsolution(-g/d,y,v,1); #6; pdsolve(y(p_,t_)+b_*y(q_,x_)+c_*y(q_,u_)+d_*y(q_,v_)+y_+f_,y_):= psolution(b,q,c,q,d,q,1,f,y,t,x,u,v,p) +gsolution(-1,y,t,p)+gsolution(-1/b,y,x,q)+gsolution(-1/c,y,u,q)+gsolution(-1/d,y,v,q); pdsolve(y(p_,t_)+b_*y(q_,x_)+c_*y(q_,u_)+d_*y(q_,v_)+g_*y_+f_,y_):= psolution(b,q,c,q,d,q,g,f,y,t,x,u,v,p) +gsolution(-g,y,t,p)+gsolution(-g/b,y,x,q)+gsolution(-g/c,y,u,q)+gsolution(-g/d,y,v,q); #0; pdsolve(a_=b_,y_):=pdsolve(replace(toy(a-b),y(x),y),y); #pdsolve(y(p_,t)=b_,y_):=if(hasnot(b,y) and hasnot(b,t),if(p>1,dsolve(toy(b),y)+c_1*t,dsolve(toy(b)-1,y)+t^p/p!),pde(y(p,t)-toy(b),y) ); #pdsolve(ds(y,t,p_)=b_,y):=if(hasnot(b,y) and hasnot(b,t),if(p>1,dsolve(toy(b),y)+c_1*t,dsolve(toy(b)-1,y)+t^p/p!),pde(y(p,t)-toy(b),y) ); pdsolve(a_):=pdsolve(replace(toy(a),y(x),y),y); pdsolve(a_=b_):=pdsolve(toy(a-b),y);