23.2.244 problem 250
Internal
problem
ID
[5599]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
250
Date
solved
:
Tuesday, September 30, 2025 at 01:10:51 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-3 a^{2} x y y^{\prime }-a^{2} x^{2}+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.043 (sec). Leaf size: 198
ode:=(-a^2+1)*y(x)^2*diff(y(x),x)^2-3*a^2*x*y(x)*diff(y(x),x)-a^2*x^2+y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (2 a^{2} \textit {\_a}^{2}-2 \textit {\_a}^{2}+3 a^{2}+\sqrt {4 a^{2} \textit {\_a}^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+3 a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (2 a^{2} \textit {\_a}^{2}-2 \textit {\_a}^{2}+3 a^{2}-\sqrt {4 a^{2} \textit {\_a}^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+3 a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 0.801 (sec). Leaf size: 342
ode=(1-a^2)y[x]^2 (D[y[x],x])^2 -2 a^2 x y[x] D[y[x],x]-a^2 x y[x] D[y[x],x]-a^2 x^2+y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solve}\left [\frac {\log \left (-\left (a^2 \left (\frac {2 y(x)^2}{x^2}+3\right )\right )+\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}+\frac {2 y(x)^2}{x^2}\right )-\frac {2 \arctan \left (\frac {1-\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}}{\sqrt {-5 a^4+2 a^2-1}}\right )}{\sqrt {-5 a^4+2 a^2-1}}}{4 a^2-4}=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\log \left (a^2 \left (\frac {2 y(x)^2}{x^2}+3\right )+\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}-\frac {2 y(x)^2}{x^2}\right )-\frac {2 \arctan \left (\frac {\sqrt {5 a^4+4 a^2 \left (\frac {y(x)^2}{x^2}+1\right )-\frac {4 y(x)^2}{x^2}}+1}{\sqrt {-5 a^4+2 a^2-1}}\right )}{\sqrt {-5 a^4+2 a^2-1}}}{4 a^2-4}=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*x**2 - 3*a**2*x*y(x)*Derivative(y(x), x) + (1 - a**2)*y(x)**2*Derivative(y(x), x)**2 + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out