8.21.14 problem 14

Internal problem ID [2723]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 05:50:14 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-y(t) = t+sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_4 \,{\mathrm e}^{-t}+\frac {\left (t +4 c_1 \right ) \cos \left (t \right )}{4}+\frac {\left (4 c_3 -1\right ) \sin \left (t \right )}{4}+c_2 \,{\mathrm e}^{t}-t \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 43

ode=D[y[t],{t,4}]-y[t]==t+Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -t+c_1 e^t+c_3 e^{-t}+\left (\frac {t}{4}+c_2\right ) \cos (t)+\left (-\frac {1}{4}+c_4\right ) \sin (t) \end{align*}

✓ Sympy. Time used: 0.084 (sec). Leaf size: 27

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

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