54.2.18 problem 594

Internal problem ID [11892]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 594
Date solved : Sunday, October 12, 2025 at 02:02:23 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 95

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

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

✓ Mathematica. Time used: 0.165 (sec). Leaf size: 236

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

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

✗ Sympy

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

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