29.16.16 problem 459

Internal problem ID [5057]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 16
Problem number : 459
Date solved : Monday, January 27, 2025 at 10:06:19 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} \left (y-\cot \left (x \right ) \csc \left (x \right )\right ) y^{\prime }+\csc \left (x \right ) \left (1+y \cos \left (x \right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 37

dsolve((y(x)-cot(x)*csc(x))*diff(y(x),x)+csc(x)*(1+y(x)*cos(x))*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \csc \left (x \right ) \left (-\sqrt {c_{1} +\cot \left (x \right )^{2}}+\cot \left (x \right )\right ) \\ y \left (x \right ) &= \csc \left (x \right ) \left (\sqrt {c_{1} +\cot \left (x \right )^{2}}+\cot \left (x \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.484 (sec). Leaf size: 85

DSolve[(y[x]-Cot[x] Csc[x])D[y[x],x]+Csc[x](1+y[x] Cos[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \cot (x) \csc (x)-\frac {i \csc ^2(x) \sqrt {(-1+c_1) \cos (2 x)-1-c_1}}{\sqrt {2}} \\ y(x)\to \cot (x) \csc (x)+\frac {i \csc ^2(x) \sqrt {(-1+c_1) \cos (2 x)-1-c_1}}{\sqrt {2}} \\ \end{align*}