62.17.7 problem Ex 7

Internal problem ID [12907]
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
Date solved : Tuesday, January 28, 2025 at 04:43:00 AM
CAS classification : [_separable]

\begin{align*} {y^{\prime }}^{2}+2 y^{\prime } y \cot \left (x \right )&=y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.188 (sec). Leaf size: 39

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 &= 0 \\ y &= \frac {\operatorname {csgn}\left (\sin \left (x \right )\right ) c_{1}}{\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )} \\ y &= \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: 0.142 (sec). Leaf size: 36

DSolve[(D[y[x],x])^2+2*D[y[x],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*}