2.11.31 problem 54

Internal problem ID [899]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 54
Date solved : Tuesday, September 30, 2025 at 04:19:14 AM
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: 0.015 (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: 0.247 (sec). Leaf size: 31

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

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