Internal problem ID [7953]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 373.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+a^{2} y^{2} \left (\ln \relax (y)^{2}-1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
dsolve(diff(y(x),x)^2+a^2*y(x)^2*(ln(y(x))^2-1) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = {\mathrm e}^{\RootOf \left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )} \\ y \relax (x ) = {\mathrm e}^{-\sin \left (-x a +a c_{1}\right )} \\ y \relax (x ) = {\mathrm e}^{\sin \left (-x a +a c_{1}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 11.45 (sec). Leaf size: 157
DSolve[a^2*(-1 + Log[y[x]]^2)*y[x]^2 + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{\cosh (\sin (a x+i c_1))+\sinh \left (\sqrt {\sin ^2(a x+i c_1)}\right )} \\ y(x)\to \cosh (\sin (a x+i c_1))+\sinh \left (\sqrt {\sin ^2(a x+i c_1)}\right ) \\ y(x)\to \cosh (\sin (a x-i c_1))-\sinh \left (\sqrt {\sin ^2(a x-i c_1)}\right ) \\ y(x)\to \cosh (\sin (a x-i c_1))+\sinh \left (\sqrt {\sin ^2(a x-i c_1)}\right ) \\ y(x)\to \frac {1}{e} \\ y(x)\to e \\ \end{align*}