66.1.52 problem Problem 66

Internal problem ID [13828]
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 : Monday, March 31, 2025 at 08:14:28 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.066 (sec). Leaf size: 39

ode:=diff(y(x),x)^2+2*y(x)*diff(y(x),x)*cot(x)-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {c_1 \,\operatorname {csgn}\left (\sin \left (x \right )\right )}{\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )} \\ y &= c_1 \left (\cos \left (x \right )+\operatorname {csgn}\left (\sec \left (x \right )\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right ) \csc \left (x \right )^{2} \\ \end{align*}

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 36

ode=D[y[x],x]^2+2*y[x]*D[y[x],x]*Cot[x]-y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 7.288 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2 + 2*y(x)*Derivative(y(x), x)/tan(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1} e^{\int \frac {\sqrt {\frac {1}{\cos ^{2}{\left (x \right )}}}}{\tan {\left (x \right )}}\, dx}}{\sin {\left (x \right )}}, \ y{\left (x \right )} = \frac {C_{1} e^{- \int \frac {\sqrt {\frac {1}{\cos ^{2}{\left (x \right )}}}}{\tan {\left (x \right )}}\, dx}}{\sin {\left (x \right )}}\right ] \]