60.1.403 problem 414
Internal
problem
ID
[10417]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
414
Date
solved
:
Sunday, March 30, 2025 at 04:41:09 PM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} x {y^{\prime }}^{2}+y y^{\prime }+x^{3}&=0 \end{align*}
✓ Maple. Time used: 0.085 (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: 1.092 (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) cannot be solved by the factorable group method