89.24.10 problem 10

Internal problem ID [24847]
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 171
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:48:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.229 (sec). Leaf size: 31

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

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