54.1.403 problem 414

Internal problem ID [11717]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 414
Date solved : Tuesday, September 30, 2025 at 10:14:16 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x {y^{\prime }}^{2}+y y^{\prime }+x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 270

ode:=x*diff(y(x),x)^2+y(x)*diff(y(x),x)+x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} -\int _{\textit {\_b}}^{x}\frac {-y+\sqrt {-4 \textit {\_a}^{4}+y^{2}}}{\left (-5 y+\sqrt {-4 \textit {\_a}^{4}+y^{2}}\right ) \textit {\_a}}d \textit {\_a} -2 \int _{}^{y}\frac {1+\left (40 \textit {\_f} -8 \sqrt {-4 x^{4}+\textit {\_f}^{2}}\right ) \int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (-5 \textit {\_f} +\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}\right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a}}{5 \textit {\_f} -\sqrt {-4 x^{4}+\textit {\_f}^{2}}}d \textit {\_f} +c_1 &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {y+\sqrt {-4 \textit {\_a}^{4}+y^{2}}}{\left (\sqrt {-4 \textit {\_a}^{4}+y^{2}}+5 y\right ) \textit {\_a}}d \textit {\_a} +2 \int _{}^{y}\frac {-1+8 \left (\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right ) \int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}}{\left (\sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}+5 \textit {\_f} \right )^{2} \sqrt {-4 \textit {\_a}^{4}+\textit {\_f}^{2}}}d \textit {\_a}}{\sqrt {-4 x^{4}+\textit {\_f}^{2}}+5 \textit {\_f}}d \textit {\_f} +c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.479 (sec). Leaf size: 107

ode=x^3 + y[x]*D[y[x],x] + x*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{5 K[2]+\sqrt {K[2]^2-4}}dK[2]\&\right ]\left [\int _1^x-\frac {1}{2 K[3]}dK[3]+c_1\right ]\\ y(x)&\to x^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {K[4]^2-4}-5 K[4]}dK[4]\&\right ]\left [\int _1^x\frac {1}{2 K[5]}dK[5]+c_1\right ] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-4*x**4 + y(x)**2) - y(x))/(2*x) can