8.21.10 problem 10

Internal problem ID [2719]
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 : 10
Date solved : Tuesday, September 30, 2025 at 05:50:11 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 71

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

\[ y = \frac {\int \sin \left (t \right ) g \left (t \right )d t \cos \left (t \right )}{2}+\frac {\int {\mathrm e}^{-t} g \left (t \right )d t {\mathrm e}^{t}}{4}-\frac {\int \cos \left (t \right ) g \left (t \right )d t \sin \left (t \right )}{2}-\frac {\int {\mathrm e}^{t} g \left (t \right )d t {\mathrm e}^{-t}}{4}+c_1 \cos \left (t \right )+c_2 \,{\mathrm e}^{t}+c_3 \sin \left (t \right )+c_4 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 118

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

\begin{align*} y(t)&\to e^t \int _1^t\frac {1}{4} e^{-K[1]} g(K[1])dK[1]+e^{-t} \int _1^t-\frac {1}{4} e^{K[3]} g(K[3])dK[3]+\sin (t) \int _1^t-\frac {1}{2} \cos (K[4]) g(K[4])dK[4]+\cos (t) \int _1^t\frac {1}{2} g(K[2]) \sin (K[2])dK[2]+c_1 e^t+c_3 e^{-t}+c_2 \cos (t)+c_4 \sin (t) \end{align*}

✓ Sympy. Time used: 0.834 (sec). Leaf size: 63

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

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