66.2.14 problem Problem 14

Internal problem ID [13842]
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 14
Date solved : Monday, March 31, 2025 at 08:14:54 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime \prime \prime }-2 x^{\prime \prime }+x&=t^{2}-3 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=diff(diff(diff(diff(x(t),t),t),t),t)-2*diff(diff(x(t),t),t)+x(t) = t^2-3; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \left (c_3 t +c_1 \right ) {\mathrm e}^{t}+\left (c_4 t +c_2 \right ) {\mathrm e}^{-t}+t^{2}+1 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode=D[x[t],{t,4}]-2*D[x[t],{t,2}]+x[t]==t^2-3; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to t^2+c_2 e^{-t} t+c_1 e^{-t}+e^t (c_4 t+c_3)+1 \]

✓ Sympy. Time used: 0.108 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**2 + x(t) - 2*Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 4)) + 3,0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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