7.11.52 problem 54

Internal problem ID [373]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 54
Date solved : Tuesday, March 04, 2025 at 11:14:32 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);
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 23

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

\[ y(x)\to \cos (x) \text {arctanh}(\cos (x))+c_1 \cos (x)+c_2 \sin (x)-1 \]

✓ Sympy. Time used: 0.311 (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)
 

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