23.1.472 problem 462

Internal problem ID [5079]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 462
Date solved : Tuesday, September 30, 2025 at 11:33:27 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*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 133

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

\begin{align*} y &= \frac {{\mathrm e}^{-i x} \csc \left (x \right )^{2} \left (\left ({\mathrm e}^{2 i x}-1\right ) \sqrt {\frac {\left (-2 c_1 +16\right ) {\mathrm e}^{2 i x}+c_1 \left ({\mathrm e}^{4 i x}+1\right )}{\left ({\mathrm e}^{i x}-1\right )^{2} \left ({\mathrm e}^{i x}+1\right )^{2}}}+2 \,{\mathrm e}^{2 i x}+2\right )}{4} \\ y &= -\frac {{\mathrm e}^{-i x} \csc \left (x \right )^{2} \left (\left ({\mathrm e}^{2 i x}-1\right ) \sqrt {\frac {\left (-2 c_1 +16\right ) {\mathrm e}^{2 i x}+c_1 \left ({\mathrm e}^{4 i x}+1\right )}{\left ({\mathrm e}^{i x}-1\right )^{2} \left ({\mathrm e}^{i x}+1\right )^{2}}}-2 \,{\mathrm e}^{2 i x}-2\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 1.057 (sec). Leaf size: 85

ode=(y[x]-Cot[x]*Csc[x])*D[y[x],x]+Csc[x]*(1+y[x]*Cos[x])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)*cos(x) + 1)*y(x)/sin(x) + (y(x) - 1/(sin(x)*tan(x)))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out