Internal problem ID [10213]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
28. Summary. Page 59
Problem number: Ex 7.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_separable]
\[ \boxed {{y^{\prime }}^{2}+2 y^{\prime } y \cot \left (x \right )-y^{2}=0} \]
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 61
dsolve(diff(y(x),x)^2+2*diff(y(x),x)*y(x)*cot(x)=y(x)^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 \\ y \left (x \right ) = \frac {c_{1} \left (\tan \left (x \right )^{2}+1\right ) \sqrt {\frac {\tan \left (x \right )^{2}}{\tan \left (x \right )^{2}+1}}}{\left (1+\sqrt {\tan \left (x \right )^{2}+1}\right ) \tan \left (x \right )} \\ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\operatorname {arctanh}\left (\frac {1}{\sqrt {\tan \left (x \right )^{2}+1}}\right )} \sqrt {\tan \left (x \right )^{2}+1}}{\tan \left (x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.155 (sec). Leaf size: 36
DSolve[(y'[x])^2+2*y'[x]*y[x]*Cot[x]==y[x]^2,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*}