14.21.10 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
:
Sunday, March 30, 2025 at 12:15:38 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: 0.055 (sec). Leaf size: 118
ode=D[y[t],{t,4}]-y[t]==g[t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to e^t \int _1^t\frac {1}{4} e^{-K[1]} g(K[1])dK[1]+e^{-t} \int _1^t-\frac {1}{4} e^{K[3]} g(K[3])dK[3]+\sin (t) \int _1^t-\frac {1}{2} \cos (K[4]) g(K[4])dK[4]+\cos (t) \int _1^t\frac {1}{2} g(K[2]) \sin (K[2])dK[2]+c_1 e^t+c_3 e^{-t}+c_2 \cos (t)+c_4 \sin (t)
\]
✓ Sympy. Time used: 1.360 (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)
\[
y{\left (t \right )} = \left (C_{1} - \frac {\int g{\left (t \right )} e^{t}\, dt}{4}\right ) e^{- t} + \left (C_{2} - \frac {\int g{\left (t \right )} \cos {\left (t \right )}\, dt}{2}\right ) \sin {\left (t \right )} + \left (C_{3} + \frac {\int g{\left (t \right )} e^{- t}\, dt}{4}\right ) e^{t} + \left (C_{4} + \frac {\int g{\left (t \right )} \sin {\left (t \right )}\, dt}{2}\right ) \cos {\left (t \right )}
\]