32.9.5 problem Exercise 22.5, page 240

Internal problem ID [5979]
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 22.5, page 240
Date solved : Wednesday, March 05, 2025 at 12:01:32 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

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

\[ y \left (x \right ) = \cos \left (x \right ) c_{1} +c_{2} \sin \left (x \right )+\frac {\cos \left (x \right )^{2}}{3}+\frac {1}{3} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 27

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

\[ y(x)\to \frac {1}{6} (\cos (2 x)+6 c_1 \cos (x)+6 c_2 \sin (x)+3) \]

✓ Sympy. Time used: 0.255 (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)
 

\[ y{\left (x \right )} = C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )} - \frac {\sin ^{2}{\left (x \right )}}{3} + \frac {2}{3} \]