60.1.491 problem 494
Internal
problem
ID
[10505]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
494
Date
solved
:
Monday, January 27, 2025 at 08:25:52 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} \left (y^{2}-a^{2} x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+\left (-a^{2}+1\right ) x^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 3.646 (sec). Leaf size: 157
dsolve((y(x)^2-a^2*x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+(-a^2+1)*x^2 = 0,y(x), singsol=all)
\begin{align*}
y &= \sqrt {a^{2}-1}\, x \\
y &= -\sqrt {a^{2}-1}\, x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}-\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}+\sqrt {a^{2} \left (\textit {\_a}^{2}-a^{2}+1\right )}+\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2}-a^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 61.655 (sec). Leaf size: 377
DSolve[(1 - a^2)*x^2 + 2*x*y[x]*D[y[x],x] + (-(a^2*x^2) + y[x]^2)*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt {-\frac {a^2 x^2+2 \sqrt {a^2 (\cosh (2 c_1)+\sinh (2 c_1)) \left (\left (a^2-1\right ) x^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}+\left (a^2+1\right ) \cosh (2 c_1)+\left (a^2+1\right ) \sinh (2 c_1)-x^2}{a^2-1}} \\
y(x)\to \sqrt {-\frac {a^2 x^2+2 \sqrt {a^2 (\cosh (2 c_1)+\sinh (2 c_1)) \left (\left (a^2-1\right ) x^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}+\left (a^2+1\right ) \cosh (2 c_1)+\left (a^2+1\right ) \sinh (2 c_1)-x^2}{a^2-1}} \\
y(x)\to -\sqrt {\frac {-a^2 x^2+2 \sqrt {a^2 (\cosh (2 c_1)+\sinh (2 c_1)) \left (\left (a^2-1\right ) x^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}-\left (a^2+1\right ) \cosh (2 c_1)-\left (a^2+1\right ) \sinh (2 c_1)+x^2}{a^2-1}} \\
y(x)\to \sqrt {\frac {-a^2 x^2+2 \sqrt {a^2 (\cosh (2 c_1)+\sinh (2 c_1)) \left (\left (a^2-1\right ) x^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}-\left (a^2+1\right ) \cosh (2 c_1)-\left (a^2+1\right ) \sinh (2 c_1)+x^2}{a^2-1}} \\
\end{align*}