Internal problem ID [8709]
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]
\[ \boxed {{y^{\prime }}^{2}+a^{2} y^{2} \left (\ln \left (y\right )^{2}-1\right )=0} \]
✓ Solution by Maple
Time used: 0.046 (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 \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (a^{2} {\mathrm e}^{2 \textit {\_Z}} \left (\textit {\_Z}^{2}-1\right )\right )} y \left (x \right ) = {\mathrm e}^{-\sin \left (a \left (-x +c_{1} \right )\right )} y \left (x \right ) = {\mathrm e}^{\sin \left (c_{1} a -a x \right )} \end{align*}
✓ Solution by Mathematica
Time used: 11.771 (sec). Leaf size: 197
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 \exp \left (-\frac {1}{2} \sqrt {-e^{2 i a x-2 c_1}-e^{2 c_1-2 i a x}+2}\right ) y(x)\to \exp \left (\frac {1}{2} \sqrt {-e^{2 i a x-2 c_1}-e^{2 c_1-2 i a x}+2}\right ) y(x)\to \exp \left (-\frac {1}{2} \sqrt {-e^{-2 i a x-2 c_1} \left (-1+e^{2 i a x+2 c_1}\right ){}^2}\right ) y(x)\to \exp \left (\frac {1}{2} \sqrt {-e^{-2 i a x-2 c_1} \left (-1+e^{2 i a x+2 c_1}\right ){}^2}\right ) y(x)\to \frac {1}{e} y(x)\to e \end{align*}