54.2.21 problem 597

Internal problem ID [11895]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 597
Date solved : Tuesday, September 30, 2025 at 11:39:09 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {2 a}{x^{2} \left (-y+2 F \left (\frac {x y^{2}-4 a}{x}\right ) a \right )} \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 55

ode:=diff(y(x),x) = 2*a/x^2/(-y(x)+2*F((x*y(x)^2-4*a)/x)*a); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\frac {\textit {\_Z}^{2} x -4 a}{x}\right )\right ) \\ -\frac {y}{2 a}+\frac {\int _{}^{y^{2}-\frac {4 a}{x}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a}}{8 a^{2}}-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.175 (sec). Leaf size: 130

ode=D[y[x],x] == (2*a)/(x^2*(2*a*F[(-4*a + x*y[x]^2)/x] - y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
F = Function("F") 
ode = Eq(-2*a/(x**2*(2*a*F((-4*a + x*y(x)**2)/x) - y(x))) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out