26.8.20 problem Exercise 21.27, page 231

Internal problem ID [7104]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 21. Undetermined Coefficients
Problem number : Exercise 21.27, page 231
Date solved : Tuesday, September 30, 2025 at 04:21:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

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

\[ y = -\frac {\sin \left (x \right )^{2} \cos \left (x \right )}{4}+\frac {\left (4 c_2 +x \right ) \sin \left (x \right )}{4}+\cos \left (x \right ) c_1 \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to \sin (x) \int _1^x\frac {1}{2} \sin ^2(2 K[1])dK[1]+c_2 \sin (x)+\cos (x) \left (-\frac {\sin ^4(x)}{2}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.229 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {\sin ^{2}{\left (x \right )}}{4}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {x}{4}\right ) \sin {\left (x \right )} \]