60.2.31 problem 607

Internal problem ID [10605]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 607
Date solved : Sunday, March 30, 2025 at 06:09:56 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x} \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 32

ode:=diff(y(x),x) = (2*y(x)+F(1/x^2*y(x))*x^3)/x; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 121

ode=D[y[x],x] == (x^3*F[y[x]/x^2] + 2*y[x])/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {K[2]}{x^2}\right ) \int _1^x\left (\frac {2}{F\left (\frac {K[2]}{K[1]^2}\right ) K[1]^3}-\frac {2 K[2] F''\left (\frac {K[2]}{K[1]^2}\right )}{F\left (\frac {K[2]}{K[1]^2}\right )^2 K[1]^5}\right )dK[1] x^2+1}{x^2 F\left (\frac {K[2]}{x^2}\right )}dK[2]+\int _1^x\left (\frac {2 y(x)}{F\left (\frac {y(x)}{K[1]^2}\right ) K[1]^3}+1\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F = Function("F") 
ode = Eq(Derivative(y(x), x) - (x**3*F(y(x)/x**2) + 2*y(x))/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3*F(y(x)/x**2) + 2*y(x))/x cannot be solved by the lie group method