66.1.52 problem Problem 66

Internal problem ID [13899]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 66
Date solved : Tuesday, January 28, 2025 at 06:07:56 AM
CAS classification : [_separable]

\begin{align*} {y^{\prime }}^{2}+2 y y^{\prime } \cot \left (x \right )-y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.209 (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 &= 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: 0.148 (sec). Leaf size: 36

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