54.2.344 problem 923

Internal problem ID [12218]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 923
Date solved : Wednesday, October 01, 2025 at 01:14:27 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{\prime }&=\frac {y^{2}+2 x y+x^{2}+{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}}{y^{2}+2 x y+x^{2}-{\mathrm e}^{-2 \left (x -y\right ) \left (x +y\right )}} \end{align*}

✓ Maple. Time used: 0.072 (sec). Leaf size: 35

ode:=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)))); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} +\int _{}^{{\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2 x \right )}\frac {1}{{\mathrm e}^{2 \textit {\_a}}+\textit {\_a}}d \textit {\_a} +c_1 \right )}-x \]

✓ Mathematica. Time used: 3.446 (sec). Leaf size: 432

ode=D[y[x],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); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 e^{2 (x-K[2]) (x+K[2])} K[2]}{-e^{2 (x-K[2]) (x+K[2])} x^2+e^{2 (x-K[2]) (x+K[2])} K[2]^2+1}-\int _1^x\left (-\frac {2 e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1] (2 (K[1]-K[2])-2 (K[1]+K[2]))}{e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1]^2-e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]^2-1}+\frac {2 e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1] \left (e^{2 (K[1]-K[2]) (K[1]+K[2])} (2 (K[1]-K[2])-2 (K[1]+K[2])) K[1]^2-2 e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]-e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]^2 (2 (K[1]-K[2])-2 (K[1]+K[2]))\right )}{\left (e^{2 (K[1]-K[2]) (K[1]+K[2])} K[1]^2-e^{2 (K[1]-K[2]) (K[1]+K[2])} K[2]^2-1\right )^2}-\frac {1}{(K[1]+K[2])^2}\right )dK[1]+\frac {1}{x+K[2]}\right )dK[2]+\int _1^x\left (\frac {1}{K[1]+y(x)}-\frac {2 e^{2 (K[1]-y(x)) (K[1]+y(x))} K[1]}{e^{2 (K[1]-y(x)) (K[1]+y(x))} K[1]^2-e^{2 (K[1]-y(x)) (K[1]+y(x))} y(x)^2-1}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**2 + 2*x*y(x) + y(x)**2 + exp((-2*x + 2*y(x))*(x + y(x))))/(x**2 + 2*x*y(x) + y(x)**2 - exp((-2*x + 2*y(x))*(x + y(x)))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out