66.2.2 problem Problem 2

Internal problem ID [13830]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 2
Date solved : Monday, March 31, 2025 at 08:14:32 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

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

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

\[ x = \frac {\cos \left (2 t \right )}{3}+\frac {\left (-t +2 c_1 \right ) \cos \left (t \right )}{2}+\frac {\left (1+4 c_2 \right ) \sin \left (t \right )}{4} \]

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 30

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

\[ x(t)\to \frac {1}{3} \cos (2 t)+\left (-\frac {t}{2}+c_1\right ) \cos (t)+c_2 \sin (t) \]

✓ Sympy. Time used: 0.107 (sec). Leaf size: 22

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

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