66.2.31 problem Problem 42

Internal problem ID [13930]
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 42
Date solved : Tuesday, January 28, 2025 at 06:08:59 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 33

dsolve(diff(x(t),t$4)+2*diff(x(t),t$2)+x(t)=cos(t),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (8 c_{3} t -t^{2}+8 c_{1} +2\right ) \cos \left (t \right )}{8}+\sin \left (t \right ) \left (\left (c_4 +\frac {3}{8}\right ) t +c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 43

DSolve[D[x[t],{t,4}]+2*D[x[t],{t,2}]+x[t]==Cos[t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \left (-\frac {t^2}{8}+c_2 t+\frac {5}{16}+c_1\right ) \cos (t)+\frac {1}{4} (t+4 c_4 t+4 c_3) \sin (t) \]