88.18.3 problem 3

Internal problem ID [24122]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 133
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:00:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

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

\[ y = \sin \left (x \right ) c_2 +\cos \left (x \right ) c_1 -1-\ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to \cos (x) \text {arctanh}(\cos (x))+c_1 \cos (x)+c_2 \sin (x)-1 \end{align*}

✓ Sympy. Time used: 0.194 (sec). Leaf size: 31

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

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