26.8.13 problem Exercise 21.16, page 231

Internal problem ID [7097]
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 21.16, page 231
Date solved : Tuesday, September 30, 2025 at 04:21:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=x \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)+4*y(x) = x*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-x^{2}+8 c_1 \right ) \cos \left (2 x \right )}{8}+\frac {\sin \left (2 x \right ) \left (x +16 c_2 \right )}{16} \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 68

ode=D[y[x],{x,2}]+4*y[x]==x*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.082 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sin(2*x) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {x^{2}}{8}\right ) \cos {\left (2 x \right )} + \left (C_{2} + \frac {x}{16}\right ) \sin {\left (2 x \right )} \]