#	differential equation solver;
#	solve differential equation and fractional differential equation for unknown y with independent variable x;
#	dsolve(eq,y)
#	there are 3 way to input derivative y: y', y(1,x), or ds(y,x,1)
#	there are 3 way to input second order derivative y: y'', y(2,x), or ds(y,x,2)
#	e.g. dsolve( y' = x*y, y)
#	dsolve( y(0.5,x)=y^2 )
#	dsolve(a_,y_,x_,q_) = If(isfree(a,y),d(a,x,-q), If(isfree(a/y,y),mittag(q,d(a/y,x,-q)), If(isfree(a,x), solve(d(1/a, y,-q)=x^q/q!,y) )));
#	dsolve(a_+b_, y_,x_,q_) := If(isfree(a,b,y), d(a+b,x,-q)+C_1+if(q>1, C_2*x,0), 
	If(isfree(a,b,x), solve(d(1/(a+b),y,-q)=x^q/q!,y)+if(q>1, C_2*x^2+C_1*x,C_1+x), 
	If(isfree(a,y) and isfree(d(b,y),y), 	block(dsolve:=d(d(b,y),x,-q),simplify(mittag(q,dsolve)*d(a*mittag(q,-dsolve),x,-q))+mittag(q,dsolve)), 
	If(isfree(b,y) and isfree(d(a,y),y), 	block(dsolve:=d(d(a,y),x,-q),simplify(mittag(q,dsolve)*d(b*mittag(q,-dsolve),x,-q))+mittag(q,dsolve)) ))));


dsolve(a_+y, y_,x_,q_) := If(isfree(a,y), If(isfree(a,x), If(q>0, C_1*exp(x)-a, C_1*mittag(-q,x^(-q)/(-q)!) ), 
	C_1*exp(x)+psolution(1,a,y,x,q) ));
#dsolve(a_+b_*y, y_,x_,q_) := If(isfree(a,b,y), if(isfree(a,x), If(b>0, C_1*exp(b^(1/q)*x)-a/b,C_1*mittag(abs(q),b^sgn(q)*x^abs(q)/(abs(q))!))-a/b, 
	If(b>0, C_1*exp(b^(1/q)*x)+psolution(b,a,y,x,q) ,C_1*mittag(abs(q),d(b,x,-q)) +psolution(b,a,y,x,q) )));

dsolve(a_+b_*y, y_,x_,q_) := If(isfree(a,b,y), If(b>0, C_1*exp(b^(1/q)*x)+if(isfree(a,x),-a/b,psolution(b,a,y,x,q)) ,C_1*mittag(abs(q),d(b,x,-q))+psolution(b,a,y,x,q) ));

#dsolve(a_+a_*y, y_,x_,q_) := If(isfree(a,y), If(isfree(a,x), If(q>0, C_1*exp(a^(1/q)*x)-1, C_1*mittag(-q,1/a*x^(-q)/(-q)!)  ), 
	If(q>0,C_1*mittag(q,d(a,x,-q))-1, C_1*mittag(-q,d(a,x,-q))*d(a,x,-q) ) ));

#dsolve(x_^q_+y_, y_,x_,q_) := -x^q-q!;
#dsolve(x_^n_+b_*y_, y_,x_,q_) := If(isfree(b,x) and isfree(b,y), If(n==q, -x^q/b-q!/b^2, If(q>0, n!*x^(n+q)*mittag(q,1+n+q,b*x^q), gamma(n+1)*x^n*mittag(-q,n+1,1/b*x^(-q)) )));
#dsolve(x_^n_+b_*y_, y_,x_,q_) := If(isfree(b,x) and isfree(b,y), If(n==q, -x^n/b-q!/b^2, n!*x^(n+q)*mittag(q,1+n+q,b*x^q) ));
#dsolve(a_*x_^n_+b_*y_, y_,x_,q_) := If(isfree(a,b,x) and isfree(a,b,y), If(n==q, -a*x^n/b-a*q!/b^2, a*n!*x^(n+q)*mittag(q,1+n+q,b*x^q) ));

#dsolve(x_^n_+y, y_,x_,q_) := If(n==q, -x^n-n!, n!*x^(n+q)*mittag(q,1+n+q,x^q));
#dsolve(a_*x_^n_+y, y_,x_,q_) := If(isfree(a,x) and isfree(a,y), If(n==q, -a*x^n-a*n!, a*n!*x^(n+q)*mittag(q,1+n+q,x^q) ));
#dsolve(x_^n_+y+c_, y_,x_,q_) := If(isfree(c,x) and isfree(c,y), If(n==q, -x^n-n!-c, n!*x^(n+q)*mittag(q,1+n+q,x^q)-c));
#dsolve(a_*x_^n_+y+c_, y_,x_,q_) := If(isfree(a,c,x) and isfree(a,c,y), If(n==q, -a*x^n-a*n!-c, a*n!*x^(n+q)*mittag(q,1+n+q,x^q)-c ));
#dsolve(x_^n_+b_*y_+c_, y_,x_,q_) := If(isfree(b,c,x) and isfree(b,c,y), If(n==q, -x^n/b-q!/b^2-c/b, n!*x^(n+q)*mittag(q,1+n+q,b*x^q)-c/b ));
#dsolve(a_*x_^n_+b_*y_+c_, y_,x_,q_) := If(isfree(a,b,c,x) and isfree(a,b,c,y), If(n==q, -a*x^n/b-a*q!/b^2-c/b, a*n!*x^(n+q)*mittag(q,1+n+q,b*x^q)-c/b ));

#dsolve(a_*x_+b_*y_, y_,x_,q_) := If(isfree(a,b,x) and isfree(a,b,y), a*x^(1+q)*mittag(q,2+q,b*x^q));
#dsolve(a_*x_+y_, y_,x_,q_) := If(isfree(a,x) and isfree(a,y), a*x^(1+q)*mittag(q,2+q,x^q));
#dsolve(x_+b_*y_, y_,x_,q_) := If(isfree(b,x) and isfree(b,y), x^(1+q)*mittag(q,2+q,b*x^q) );
#dsolve(x_+y, y_,x_,q_) :=  x^(1+q)*mittag(q,2+q,x^q);
#dsolve(a_*x_+y_+c_, y_,x_,q_) := If(isfree(a,c,x) and isfree(a,c,y), If(q>0,a*x^(1+q)*mittag(q,2+q,x^q)-c,a*x^(1+q)*mittag(q,2+q,x^q) ));
#dsolve(x_+b_*y_+c_, y_,x_,q_) := If(isfree(b,c,x) and isfree(c,b,y), If(q>0,x^(1+q)*mittag(q,2+q,b*x^q)-c/b,x^(1+q)*mittag(q,2+q,b*x^q) ));
#dsolve(x_+y_+c_, y_,x_,q_) := If(isfree(c,x) and isfree(c,y), If(q>0,x^(1+q)*mittag(q,2+q,x^q)-c,x^(1+q)*mittag(q,2+q,x^q) ));
#dsolve(a_*x_+b_*y_+c_, y_,x_,q_) := If(isfree(a,b,c,x) and isfree(a,b,c,y), If(q>0, a*x^(1+q)*mittag(q,2+q,b*x^q)-c/b,a*x^(1+q)*mittag(q,2+q,b*x^q) ));

#dsolve(a_*x_+x_*y_, y_,x_,q_) := If(isfree(a,x) and isfree(a,y), C_1*mittag(q,x^(1+q)/(1+q)!)-a);
#dsolve(x_+b_*x_*y_, y_,x_,q_) := if(isfree(b,x) and isfree(a,y), C_1*mittag(abs(q),b*x^(1+abs(q))/(1+abs(q))!)-1/b);
dsolve(a_*x_+b_*x_*y(p_,x_), y_,x_,q_) := If(isfree(a,b,x) and isfree(a,b,y), C_1*d(mittag(q-p,b*x^(1+q-p)/(1+q-p)!)-a/b,x,-p));

dsolve(b_*y^n_, y_,x_,q_) := If(isfree(b,y), If(isfree(b,x), b^(1-n)*fallingfactorial(q/(1-n),q)^(1/(n-1))*(x+C_1)^(q/(1-n)), (1/fallingfactorial(-n,q)*d(b,x,-q))^(1/(q-n)) ));
dsolve(y_^n_, y_,x_,q_) := fallingfactorial(q/(1-n),q)^(1/(n-1))*(x+C_1)^(q/(1-n)) ;

dsolve(b_*y_, y_,x_,q_) := If(b>0, C_1*exp(b^(1/q)*x)+if(iseven(q),C_2*exp(-b^(1/q)*x),0), If(b<0, C_1*mittag(q,b*x^q/q!), C_1*mittag(q,d(b,x,-q)) ));
dsolve(b_*x_*y_, y_,x_,q_) := If(b>0, C_1*exp(b^(1/q)*q/(1+q)*x^(1/q+1)), If(b<0, If(q>0, C_1*mittag(q,b*x^(q+1)/(1+q)!), C_1*mittag(-q,1/b*x^(1-q)) )));
dsolve(x_*y_, y_,x_,q_) := If(q>0, C_1*exp(q/(1+q)*x^(1/q+1)), C_1*mittag(-q,x^(1-q)/(1-q)!));
dsolve(y_, y_,x_,q_) := if(q==3, C_1*exp(x)+C_2*cos(x)+C_3*sin(x), if(iseven(q),C_1*exp(-x)+C_2*exp(x), C_1*exp(x) ));
dsolve(e^x_, y_,x_,q_) := exp(x)+if(q>0, sum(C(k+1)*x^k,k,0,ceil(q)-1,1),0 );
dsolve(0, y_,x_,q_) := if(q>0, sum(C(k+1)*x^k,k,0,ceil(q)-1,1) );

dsolve(f_, y_,x_,1) :=dsolve(f, y,x);

dsolve(y(p_,x_),y_,x_,q_):= C_1*e^x+if(p>0,C_2,0);
dsolve(b_*y(p_,x_),y_,x_,q_):=if(isfree(b,y), if(b>0, C_1*exp(b^(1/(q-p))*x)+if(p>0,C_2,0), C_1*d(mittag(q-p,d(b,x,p-q)),x,-p)+if(p>0,C_2,0) ));
dsolve(y(p_,x_)+c_,y_,x_,q_):= d(dsolve(c+y,y,x,q-p),x,-p);
dsolve(b_*y(p_,x_)+c_,y_,x_,q_):=if(isfree(b,y), d(dsolve(c+b*y,y,x,q-p),x,-p));
dsolve(y(p_,x_)+c_*y, y_,x_,q_) := if(isfree(c,y) and isfree(c,x), if(q==2p, C_1*e^(((1-sqrt(1+4c))/2)^(1/p)*x)+C_2*e^(((1+sqrt(1+4c))/2)^(1/p)*x),C_1*exp(nsolve(y^q-y^p-c,y)*x) ));
dsolve(y(p_,x_)+y, y_,x_,q_) := if(q==2p, C_1*e^(((1-sqrt(5))/2)^(1/p)*x)+C_2*e^(((1+sqrt(5))/2)^(1/p)*x),C_1*exp(nsolve(y^q-y^p-1,y)*x) );
dsolve(b_*y(p_,x_)+c_*y, y_,x_,q_) := if(isfree(b,c,y) and isfree(b,c,x), if(q==2p, if(b*b+4c==0, C_1*exp((b/2)^(1/p)*x)+C_2*exp((b/2)^(1/p)*x)*x, 
	C_1*e^(((b-sqrt(b^2+4c))/2)^(1/p)*x)+C_2*e^(((b+sqrt(b^2+4c))/2)^(1/p)*x)), 
	if(b+c==1, C_1*e^(x),C_1*exp(nsolve(y^q-b*y^p-c,y)*x) )));
dsolve(b_*y(p_,x_)+y, y_,x_,q_) := if(isfree(b,y) and isfree(b,x), if(q==2p, C_1*e^(((b-sqrt(b^2+4))/2)^(1/p)*x)+C_2*e^(((b+sqrt(b^2+4))/2)^(1/p)*x),C_1*exp(nsolve(y^q-b*y^p-1,y)*x) ));

dsolve(b_*y(p_,x_)+c_*y+d_, y_,x_,q_) := if(isfree(b,c,d,y) and isfree(b,c,d,x), dsolve(b*y(p,x)+c*y, y,x,q)-d/c, dsolve(b*y(p,x)+c*y, y,x,q)+psolution(b,p,c,d,y,x,q) );
dsolve(b_*y(p_,x_)+y+d_, y_,x_,q_) := if(isfree(b,d,y) and isfree(b,d,x), dsolve(b*y(p,x)+y, y,x,q)-d, dsolve(b*y(p,x)+y, y,x,q)+psolution(b,p,1,d,y,x,q) );
dsolve(y(p_,x_)+c_*y+d_, y_,x_,q_) := if(isfree(c,d,y) and isfree(c,d,x), dsolve(y(p,x)+c*y, y,x,q)-d/c, dsolve(y(p,x)+c*y, y,x,q)+psolution(1,p,c,d,y,x,q) );
dsolve(y(p_,x_)+y+d_, y_,x_,q_) := if(isfree(d,y) and isfree(d,x), dsolve(y(p,x)+y, y,x,q)-d, dsolve(y(p,x)+y, y,x,q)+psolution(1,p,1,d,y,x,q) );


dsolve(y(1,x_),y_,x_,3):=C_1+C_2*e^(x)+C_3*e^(-x);
dsolve(y(2,x_),y_,x_,3):=C_1+C_2*e^(x)+C_3*x;
dsolve(b_*y(1,x_),y_,x_,3):=gsolution(0,b,y,x,2)+C_3;
dsolve(b_*y(2,x_),y_,x_,3):=C_1+C_2*e^(int(b,x))+C_3*x;

dsolve(a_+b_*y_, y_,x_,2) := if(isfree(a,b,y), if(isfree(a,b,x), gsolution(0,b,y,x,2)-a/b, if(a==b, gsolution(0,b,y,x,2), gsolution(0,b,y,x,2)+psolution(b,a,y,x,2) )));
dsolve(a_+y_, y_,x_,2) := if(isfree(a,y),if(isfree(a,x), C_1*exp(x)+C_2*exp(-x)-a, C_1*exp(x)+C_2*exp(-x)-1/2*exp(-x)*int(exp(x)*a,x)+1/2*exp(x)*int(exp(-x)*a,x) ));
dsolve(x_+y_, y_,x_,2) := C_1*exp(x)+C_2*exp(-x)-x;
dsolve(x_+b_*y_, y_,x_,2) := if(isfree(b,x), gsolution(0,b,y,x,2)-x/b);
dsolve(a_*x_+b_*y_, y_,x_,2) := if(isfree(a,b,x), gsolution(0,b,y,x,2)-a*x/b);
dsolve(a_*x_+b_*y_+c_, y_,x_,2) := if(isfree(a,b,c,x), gsolution(0,b,y,x,2)-a*x/b-c/b, gsolution(0,b,y,x,2)-a*x/b+psolution(b,c,y,x,2) );

dsolve(b_*y_, y_,x_,2) := if(isfree(b,y), gsolution(0,b,y,x,2) );
dsolve(x_*y_, y_,x_,2) := C_1*AiryAi(x)+C_2*AiryBi(x);
dsolve(a_*x_*y_, y_,x_,2) := if(isfree(a,x) and isfree(a,y), C_1*AiryAi(cbrt(a)*x)+C_2*AiryBi(cbrt(a)*x));
dsolve(b_*y(1,x_),y_,x_,2):=if(isfree(b,y),if(isfree(b,x), C_1+C_2*e^(b*x), C_2*integrate(exp(int(b,x)),x)+C_1 ));
dsolve(xy_*y(1,x_),xy_,x_,2):= sqrt(2)*tan(sqrt(2)/2*x+C_1);
dsolve(b_*xy_*y(1,x_),xy_,x_,2):=if(isfree(b,xy) and isfree(b,x), sqrt(2)* sqrt(C_1)* tan(1/2 *(sqrt(2)* sqrt(b)* sqrt(C_1)* x + sqrt(2)* sqrt(b)* sqrt(C_1))) );
dsolve(-y*y(1,x_),y_,x_,2):= sqrt(2)*tanh(sqrt(2)/2*x+C_1);
dsolve(y(1,x_)^n_,y_,x_,2):= if(n==2, C_2 - log(C_1 + x),C_2 - (-(n - 1)* (C_1 + x))^(1/(1 - n) + 1)/(n - 2));
dsolve(b_*y(1,x_)^n_,y_,x_,2):= if(isfree(b,y) and isfree(b,x),if(n==2, C_2 - log(C_1 + b*x)/b,C_2 - (-(n - 1)* (C_1 + b*x))^(1/(1 - n) + 1)/(n - 2)/b ));

dsolve(c_+y(1,x_),y_,x_,2):=if(isfree(c,y),if(isfree(c,x), C_1+C_2*e^x-c*x, C_1+C_2*e^(x)+integrate(exp(x)*int(exp(-x)*c,x),x) ));
#dsolve(c_+b_*y(1,x_),y_,x_,2):=if(isfree(b,c,y),if(isfree(b,c,x), C_1+C_2*e^(b*x)-c*x/b, integrate(dsolve(c+b*y,y,x),x)+C_2 ));
#dsolve(c_+b_*y(1,x_),y_,x_,2):=if(isfree(b,c,y),if(isfree(b,c,x), C_1+C_2*e^(b*x)-c*x/b, C_1+C_2*e^(b*x)+integrate(exp(b*x)*int(exp(-b*x)*c,x),x) ));
dsolve(c_+b_*y(1,x_),y_,x_,2):=if(isfree(b,c,y),if(isfree(b,c,x), C_1+C_2*e^(b*x)-c*x/b, C_2+int(C_1*e^(int(b,x))+psolution(b,c,y,x,1),x) ));
dsolve(xy+y(1,x_),xy_,x_,2):= C_1*e^((1-sqrt(5))/2*x)+C_2*e^((1+sqrt(5))/2*x);
dsolve(b_*y(1,x_)+xy_,xy_,x_,2):=if(isfree(b,xy) and isfree(b,x), gsolution(b,1,xy,x,2) );
dsolve(y(1,x_)+c_*xy_,xy_,x_,2):=if(isfree(c,xy) and isfree(c,x), gsolution(1,c,xy,x,2) );
dsolve(b_*y(1,x_)+c_*xy_,xy_,x_,2):=if(isfree(b,c,xy) and isfree(b,c,x), gsolution(b,c,xy,x,2) );

dsolve(y(1,x_)+xy_+d_,xy_,x_,2):=if(isfree(d,xy) and isfree(d,x), gsolution(1,1,xy,x,2)-d*x-d, gsolution(1,1,xy,x,2)+psolution(1,1,1,d,xy,x,2));
dsolve(b_*y(1,x_)+xy_+d_,xy_,x_,2):=if(isfree(b,d,xy) and isfree(b,d,x), gsolution(b,1,xy,x,2)-d*x-d*b, gsolution(b,1,xy,x,2) +psolution(b,1,1,d,xy,x,2));
dsolve(y(1,x_)+c_*xy_+d_,xy_,x_,2):=if(isfree(c,d,xy) and isfree(c,d,x), gsolution(1,c,xy,x,2)-d*x/c-d/c^2, gsolution(1,c,xy,x,2)+psolution(1,1,c,d,xy,x,2));
dsolve(b_*y(1,x_)+c_*xy_+d_,xy_,x_,2):=if(isfree(b,c,d,xy) and isfree(b,c,d,x), gsolution(b,c,xy,x,2)-d*x/b-d/c*b, gsolution(b,c,xy,x,2)+psolution(b,1,c,d,xy,x,2));

#dsolve(b_*y(1,x_)+c_*xy_+d_,xy_,x_,2):=if(isfree(b,c,d,xy) and isfree(b,c,d,x), gsolution(b,c,xy,x,2)-d*x/b-d/c/b, 
	if(b*b+4c==0, C_1*exp(b/2*x)+C_2*exp(b/2*x)*x+expand(exp(b/2*x)*x*int(exp(-b/2*x)*d,x)-exp(b/2*x)*int(d*exp(-b/2*x)*x,x)),
	gsolution(b,c,xy,x,2)+ expand(-e^((b-sqrt(b^2+4c))/2*x)*int(d*e^((-b+sqrt(b^2+4c))/2*x),x)+e^((b+sqrt(b^2+4c))/2*x)*int(d*e^((-b-sqrt(b^2+4c))/2*x),x)/sqrt(b*b+4c)) 
 ));

dsolve(y_, y_,x_,2) := C_1*exp(-x)+C_2*exp(x);

dsolve(a_ and b_, y_,x_,p_) :=	dsolve(a, y,x,p) and dsolve(b, y,x,p);
dsolve(a_ and b_, y_,x_) :=	dsolve(a, y,x) and dsolve(b, y,x);

#dsolve(a_+b_, y_,x_) := If(isfree(a,b,y), integrate(a+b,x)+C_1, 
	If(has(a,y) and has(b,y) and isfree(d(a+b,y),y), exp(integrate(d(a+b,y),x))*C_1, 
	If(isfree(b,y) and isfree(d(a,y),y),  	block(dsolve:=int(d(a,y),x),exp(dsolve)*int(b*exp(-dsolve),x)+exp(dsolve)*C_1), 
	If(isfree(a,y) and isfree(d(b,y),y), 	block(dsolve:=int(d(b,y),x),exp(dsolve)*int(a*exp(-dsolve),x)+exp(dsolve)*C_1) ))));
#dsolve(a_+b_*y_,y_,x_):=If(isfree(a,b,y), If(isfree(a,b,x), C_1*exp(b*x)-a/b,
	block(dsolve:=integrate(b,x), expand(exp(dsolve)*integrate(a*exp(-dsolve),x))+exp(dsolve)*C_1 ) ));

dsolve(a_+b_*y_,y_,x_):=If(isfree(a,b,y), If(isfree(a,b,x), C_1*exp(b*x)-a/b,	C_1*exp(integrate(b,x))+psolution(b,a,y,x,1) ));
dsolve(a_+a_*y_,y_,x_):=If(isfree(a,y),C_1*exp(integrate(a,x))-1);
dsolve(a_+y_,y_,x_):=If(isfree(a,y), If(isfree(a,x), C_1*exp(x)-a,	expand(exp(x)*integrate(a*exp(-x),x))+exp(x)*C_1  ));

dsolve(a_*y_+b_*y_^n_,y_,x_):=If(isfree(a,b,y) , block(dsolve:=integrate(a,x), exp(dsolve)*(C_1+(1-n)*integrate(b*exp((n-1)*dsolve),x))^(1/(1-n)) ) );
dsolve(a_*y_+y_^n_,y_,x_):=If(isfree(a,n,y) , block(dsolve:=integrate(a,x), exp(dsolve)*(C_1+(1-n)*integrate(exp((n-1)*dsolve),x))^(1/(1-n)) ) );
dsolve(y_+y_^n_,y_,x_):= exp(x)*(C_1-exp((n-1)*x))^(1/(1-n));
dsolve(x_+y_, y_,x_) := C_1*exp(x)-x-1;

#dsolve(a_*x_+y_, y_,x_) := If(isfree(a, x) and isfree(a, y), -a*x-a);
#dsolve(x_+b_*y_,y_,x_)  := If(isfree(b, x) and isfree(b, y), -1/b*x-1/b^2);
#dsolve(a_*x_+b_*y_,y_,x_):=If(isfree(a,b, x) and isfree(a,b, y), -a/b*x-a/b^2);
#dsolve(x_+y_+c_, y_,x_) := If(isfree(c,x) and isfree(c,y), -x-1-c);
#dsolve(x_+b_*y_+c_, y_,x_) := If(isfree(b,c,x) and isfree(b,c,y), -1/b*x-1/b^2-c/b);
#dsolve(a_*x_+y_+c_, y_,x_) := If(isfree(a,c,x) and isfree(a,c,y), -a*x-a-c);
#dsolve(a_*x_+b_*y_+c_, y_,x_) := If(isfree(a,b,c,x) and isfree(a,b,c,y), -a/b*x-a/b^2-c/b);

dsolve(b_*y_, y_,x_) := If(isfree(b,y), C_1*exp(integrate(b,x)));
dsolve(b_*y_^n_, y_,x_) := If(isfree(b,y), ((1-n)*integrate(b,x)+C_1)^(1/(1-n)) );
dsolve(y_,y_,x_) := C_1*exp(x);
#dsolve(y(n_,x_),y_,x_) := C_1*exp(x);


#dsolve(f_,y(x_)) := dsolve(replace(d(y(x),x)-f,y(x),y),y);
#dsolve(a_=b_,y_):= if(has(a,ds(y,x,2)),dsolve(ds(y,x,2)-a+b,y,x,2), if(has(a,ds(y)),dsolve(ds(y)-a+b,y,x), if(has(a,dy),dsolve(dy/dx-a+b,y,x), dsolve(a-b,y)  )));
#dsolve(y'=f_,y) := dsolve(f,y,x);
#dsolve(y''=f_,y) := dsolve(f,y,x,2);
#dsolve(dy/dx=f_,y) := dsolve(f,y,x);
#dsolve(d(y(x_),x_)=f_,y(x_)) := dsolve(replace(f,y(x),y),y,x);
#dsolve(ds(y_)=a_,y_) := dsolve(a,y,x);
#dsolve(ds(y_,x_)=a_,y_) := dsolve(a,y,x);
#dsolve(ds(y_,x_,q_)=a_,y_):= dsolve(a,y,x,q);
#dsolve(a_+ds(y_,x_,q_),y_):= dsolve(-a,y,x,q);
#dsolve(ds(y_,x_,q_)+z_,y_):= dsolve(-z,y,x,q);
#dsolve(ds(y_,x_,q_)+f_+z_,y_):= dsolve(-f-z,y,x,q);
#dsolve(a_+ds(y_,x_,q_)+z_,y_):= dsolve(-a-z,y,x,q);

dsolve(a_+y(q_,x_),y_):= dsolve(-a,y,x,q);
dsolve(a_+b_*y(q_,x_),y_):= dsolve(-expand(a/b),y,x,q);
dsolve(a_+y(q_,x_),y(x_)):= dsolve(-replace(a,y(x),y),y,x,q);
dsolve(y(p_,x_)^n_+b_,y_):= dsolve(solve(yy^n+replace(b,y(p,x),yy),yy),y,x,p);
dsolve(a_*y(p_,x_)^n_+b_,y_):= dsolve(solve(yy^n+expand(replace(b/a,y(p,x),yy)),yy),y,x,p);

dsolve(a_=b_,y_):=dsolve(toyn(a-b),y);
dsolve(y(n_,x_)=a_,y_) := dsolve(toyn(a),y,x,n);

dsolve(a_):=dsolve(replace(toyn(a),y(x),y),y);