68.1.16 problem 23

Internal problem ID [17083]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 23
Date solved : Thursday, October 02, 2025 at 01:42:49 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=t \cos \left (t \right )-\cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(diff(x(t),t),t)+x(t) = t*cos(t)-cos(t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {\left (t^{2}+4 c_2 -2 t -1\right ) \sin \left (t \right )}{4}+\frac {\cos \left (t \right ) \left (t +4 c_1 -2\right )}{4} \]

✓ Mathematica. Time used: 0.134 (sec). Leaf size: 58

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

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

✓ Sympy. Time used: 0.076 (sec). Leaf size: 24

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

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