Internal problem ID [8503]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 923.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{2}+2 y x +x^{2}+{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}}{y^{2}+2 y x +x^{2}-{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 36
dsolve(diff(y(x),x) = (y(x)^2+2*x*y(x)+x^2+exp(-2*(x-y(x))*(x+y(x))))/(y(x)^2+2*x*y(x)+x^2-exp(-2*(x-y(x))*(x+y(x)))),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\RootOf \left (-\textit {\_Z} +\int _{}^{{\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} x}\frac {1}{{\mathrm e}^{2 \textit {\_a}}+\textit {\_a}}d \textit {\_a} +c_{1}\right )}-x \]
✓ Solution by Mathematica
Time used: 2.253 (sec). Leaf size: 432
DSolve[y'[x] == (E^(-2*(x - y[x])*(x + y[x])) + x^2 + 2*x*y[x] + y[x]^2)/(-E^(-2*(x - y[x])*(x + y[x])) + x^2 + 2*x*y[x] + y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 e^{2 (x-K[2]) (x+K[2])} K[2]}{-e^{2 (x-K[2]) (x+K[2])} x^2+e^{2 (x-K[2]) (x+K[2])} K[2]^2+1}-\int _1^x\left (-\frac {2 e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1] (2 (K[1]-K[2])-2 (K[1]+K[2]))}{e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1]^2-e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]^2-1}+\frac {2 e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1] \left (e^{2 (K[1]-K[2]) (K[1]+K[2])} (2 (K[1]-K[2])-2 (K[1]+K[2])) K[1]^2-2 e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]-e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]^2 (2 (K[1]-K[2])-2 (K[1]+K[2]))\right )}{\left (e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1]^2-e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]^2-1\right )^2}-\frac {1}{(K[1]+K[2])^2}\right )dK[1]+\frac {1}{x+K[2]}\right )dK[2]+\int _1^x\left (\frac {1}{K[1]+y(x)}-\frac {2 e^{2 (K[1]-y(x)) (K[1]+y(x))} K[1]}{e^{2 (K[1]-y(x)) (K[1]+y(x))} K[1]^2-e^{2 (K[1]-y(x)) (K[1]+y(x))} y(x)^2-1}\right )dK[1]=c_1,y(x)\right ] \]