26.9.4 problem Exercise 22.4, page 240

Internal problem ID [7113]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 22. Variation of Parameters
Problem number : Exercise 22.4, page 240
Date solved : Tuesday, September 30, 2025 at 04:21:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-y(x) = sin(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} c_2 +{\mathrm e}^{x} c_1 +\frac {\cos \left (x \right )^{2}}{5}-\frac {3}{5} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 70

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

\begin{align*} y(x)&\to e^{-x} \left (e^{2 x} \int _1^x\frac {1}{2} e^{-K[1]} \sin ^2(K[1])dK[1]+\int _1^x-\frac {1}{2} e^{K[2]} \sin ^2(K[2])dK[2]+c_1 e^{2 x}+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.291 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{x} + \frac {\cos {\left (2 x \right )}}{10} - \frac {1}{2} \]