Internal problem ID [8038]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 458.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {x^{2} \left (-a^{2}+x^{2}\right ) \left (y^{\prime }\right )^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 90
dsolve(x^2*(-a^2+x^2)*diff(y(x),x)^2-1 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}}{x}\right )}{\sqrt {-a^{2}}}+c_{1} \\ y \relax (x ) = \frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}}{x}\right )}{\sqrt {-a^{2}}}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 116
DSolve[-1 + x^2*(-a^2 + x^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x \sqrt {x^2-a^2} \cot ^{-1}\left (\frac {a}{\sqrt {x^2-a^2}}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \\ y(x)\to \frac {x \sqrt {x^2-a^2} \cot ^{-1}\left (\frac {a}{\sqrt {x^2-a^2}}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \\ \end{align*}