Internal problem ID [3662]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 936.
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: 0.102 (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: 0.027 (sec). Leaf size: 116
DSolve[x^2(a^2-x^2) (y'[x])^2+1==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*}