60.1.486 problem 499
Internal
problem
ID
[10500]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
499
Date
solved
:
Sunday, March 30, 2025 at 05:20:38 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-2 a^{2} x y y^{\prime }+y^{2}-a^{2} x^{2}&=0 \end{align*}
✓ Maple. Time used: 0.144 (sec). Leaf size: 191
ode:=(-a^2+1)*y(x)^2*diff(y(x),x)^2-2*a^2*x*y(x)*diff(y(x),x)+y(x)^2-a^2*x^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {x a}{\sqrt {-a^{2}+1}} \\
y &= -\frac {x a}{\sqrt {-a^{2}+1}} \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}-\sqrt {a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}+\sqrt {a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (a^{2} \textit {\_a}^{2}-\textit {\_a}^{2}+a^{2}\right ) \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 6.193 (sec). Leaf size: 251
ode=-(a^2*x^2) + y[x]^2 - 2*a^2*x*y[x]*D[y[x],x] + (1 - a^2)*y[x]^2*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*x**2 - 2*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