60.2.44 problem 620

Internal problem ID [10618]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 620
Date solved : Sunday, March 30, 2025 at 06:10:46 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.071 (sec). Leaf size: 36

ode:=diff(y(x),x) = (y(x)^2+2*x*y(x)+x^2+exp(2*F(-(x-y(x))*(x+y(x)))))/(y(x)^2+2*x*y(x)+x^2-exp(2*F(-(x-y(x))*(x+y(x))))); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.716 (sec). Leaf size: 205

ode=D[y[x],x] == (E^(2*F[(-x + y[x])*(x + y[x])]) + x^2 + 2*x*y[x] + y[x]^2)/(-E^(2*F[(-x + y[x])*(x + y[x])]) + x^2 + 2*x*y[x] + y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 K[2]}{-x^2+e^{2 F((K[2]-x) (x+K[2]))}+K[2]^2}-\int _1^x\left (\frac {2 K[1] \left (-4 e^{2 F((K[2]-K[1]) (K[1]+K[2]))} F''((K[2]-K[1]) (K[1]+K[2])) K[2]-2 K[2]\right )}{\left (K[1]^2-e^{2 F((K[2]-K[1]) (K[1]+K[2]))}-K[2]^2\right )^2}-\frac {1}{(K[1]+K[2])^2}\right )dK[1]+\frac {1}{x+K[2]}\right )dK[2]+\int _1^x\left (\frac {1}{K[1]+y(x)}-\frac {2 K[1]}{K[1]^2-e^{2 F((y(x)-K[1]) (K[1]+y(x)))}-y(x)^2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out