20.4.36 problem Problem 52

Internal problem ID [3671]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 52
Date solved : Monday, January 27, 2025 at 07:53:50 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+y \cot \left (x \right )&=y^{3} \sin \left (x \right )^{3} \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 3.622 (sec). Leaf size: 34

dsolve([diff(y(x),x)+y(x)*cot(x)=y(x)^3*sin(x)^3,y(1/2*Pi) = 1],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\csc \left (x \right ) \sqrt {\left (2 \cos \left (x \right )-1\right )^{2} \left (1+2 \cos \left (x \right )\right )}}{1-4 \cos \left (x \right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.946 (sec). Leaf size: 20

DSolve[{D[y[x],x]+y[x]*Cot[x]==y[x]^3*Sin[x]^3,{y[Pi/2]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{\sqrt {\sin ^2(x) (2 \cos (x)+1)}} \]