66.2.34 problem Problem 49

Internal problem ID [13854]
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 : Wednesday, March 05, 2025 at 10:19:44 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

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

ode:=diff(diff(diff(diff(x(t),t),t),t),t)+x(t) = t^3; 
dsolve(ode,x(t), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 0.168 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**3 + x(t) + Derivative(x(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t^{3} + \left (C_{1} \sin {\left (\frac {\sqrt {2} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{- \frac {\sqrt {2} t}{2}} + \left (C_{3} \sin {\left (\frac {\sqrt {2} t}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{\frac {\sqrt {2} t}{2}} \]