54.2.116 problem 692

Internal problem ID [11990]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 692
Date solved : Sunday, October 12, 2025 at 02:08:45 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {y+x^{3} \sqrt {y^{2}+x^{2}}}{x} \end{align*}

✓ Maple. Time used: 1.923 (sec). Leaf size: 30

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

\[ \ln \left (\sqrt {y^{2}+x^{2}}+y\right )-\frac {x^{3}}{3}-\ln \left (x \right )-c_{1} = 0 \]

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to x \sinh \left (\frac {x^3}{3}+c_1\right ) \end{align*}

✗ Sympy

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

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