89.25.7 problem 8

Internal problem ID [24865]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Exercises at page 177
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:48:50 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+y(x) = csc(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 +\cot \left (x \right )\right ) \cos \left (x \right )+\sin \left (x \right ) c_2 -\frac {\csc \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 25

ode=D[y[x],{x,2}]+y[x]== Csc[x]^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\csc (x)}{2}+c_2 \sin (x)+\cos (x) (\cot (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.162 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - csc(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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