Internal problem ID [8793]
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]
\[ \boxed {x^{2} \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}=1} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 111
dsolve(x^2*(-a^2+x^2)*diff(y(x),x)^2-1 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {c_{1} \sqrt {-a^{2}}-\ln \left (2\right )-\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )}{\sqrt {-a^{2}}} \\ y \left (x \right ) &= \frac {c_{1} \sqrt {-a^{2}}+\ln \left (2\right )+\ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )}{\sqrt {-a^{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 120
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} \arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \\ y(x)\to \frac {x \sqrt {x^2-a^2} \arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a \sqrt {x^4-a^2 x^2}}+c_1 \\ \end{align*}