54.2.2 problem 578

Internal problem ID [11876]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 578
Date solved : Tuesday, September 30, 2025 at 11:38:24 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 100

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

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

✗ Sympy

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

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