Internal problem ID [8729]
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]
\[ \boxed {{y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2}=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 39
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 \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1}}{\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )} \\ y \left (x \right ) &= \csc \left (x \right )^{2} \left (\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.161 (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*}