Internal problem ID [7973]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 393.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+2 y y^{\prime } \cot \relax (x )-y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.318 (sec). Leaf size: 61
dsolve(diff(y(x),x)^2+2*y(x)*diff(y(x),x)*cot(x)-y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {c_{1} \left (\tan ^{2}\relax (x )+1\right ) \sqrt {\frac {\tan ^{2}\relax (x )}{\tan ^{2}\relax (x )+1}}}{\left (1+\sqrt {\tan ^{2}\relax (x )+1}\right ) \tan \relax (x )} \\ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\arctanh \left (\frac {1}{\sqrt {\tan ^{2}\relax (x )+1}}\right )} \sqrt {\tan ^{2}\relax (x )+1}}{\tan \relax (x )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 36
DSolve[-y[x]^2 + 2*Cot[x]*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \csc ^2\left (\frac {x}{2}\right ) \\ y(x)\to c_1 \sec ^2\left (\frac {x}{2}\right ) \\ y(x)\to 0 \\ \end{align*}