60.2.8 problem 584

Internal problem ID [10582]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 584
Date solved : Sunday, March 30, 2025 at 06:08:28 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 35

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

\[ \frac {y}{2 a}+\frac {\int _{}^{y^{2}-4 a x}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a}}{8 a^{2}}-c_1 = 0 \]

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 115

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

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

✗ Sympy

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

NotImplementedError : The given ODE -2*a/(2*a*F(-4*a*x + y(x)**2) + y(x)) + Derivative(y(x), x) cannot be solved by the lie group method