66.2.34 problem Problem 49

Internal problem ID [13933]
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 49
Date solved : Tuesday, January 28, 2025 at 06:09:07 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime \prime \prime }+x&=t^{3} \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 67

dsolve(diff(x(t),t$4)+x(t)=t^3,x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 78

DSolve[D[x[t],{t,4}]+x[t]==t^3,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to e^{-\frac {t}{\sqrt {2}}} \left (e^{\frac {t}{\sqrt {2}}} t^3+\left (c_1 e^{\sqrt {2} t}+c_2\right ) \cos \left (\frac {t}{\sqrt {2}}\right )+\left (c_4 e^{\sqrt {2} t}+c_3\right ) \sin \left (\frac {t}{\sqrt {2}}\right )\right ) \]