75.3.2 problem 2

Internal problem ID [19906]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 2
Date solved : Thursday, October 02, 2025 at 05:00:46 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

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

\[ y = \left (-\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )+c_1 \right ) \csc \left (x \right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 16

ode=D[y[x],x]+Cot[x]*y[x]==Csc[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \csc (x) (-\text {arctanh}(\cos (x))+c_1) \end{align*}

✓ Sympy. Time used: 0.619 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)/tan(x) + Derivative(y(x), x) - 1/sin(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} + \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{2}}{\sin {\left (x \right )}} \]