14.21.11 problem 11
Internal
problem
ID
[2720]
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
:
11
Date
solved
:
Sunday, March 30, 2025 at 12:15:39 AM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }+y&=g \left (t \right ) \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 223
ode:=diff(diff(diff(diff(y(t),t),t),t),t)+y(t) = g(t);
dsolve(ode,y(t), singsol=all);
\[
y = \frac {{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\sqrt {2}\, \int {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} g \left (t \right ) \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )d t \cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right ) {\mathrm e}^{\sqrt {2}\, t} \int g \left (t \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )d t +\sqrt {2}\, \int {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} g \left (t \right ) \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )d t \sin \left (\frac {\sqrt {2}\, t}{2}\right )-\sqrt {2}\, \cos \left (\frac {\sqrt {2}\, t}{2}\right ) {\mathrm e}^{\sqrt {2}\, t} \int g \left (t \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )d t +\left (4 c_3 \,{\mathrm e}^{\sqrt {2}\, t}+4 c_1 \right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )+4 \sin \left (\frac {\sqrt {2}\, t}{2}\right ) \left (c_4 \,{\mathrm e}^{\sqrt {2}\, t}+c_2 \right )\right )}{4}
\]
✓ Mathematica. Time used: 0.324 (sec). Leaf size: 331
ode=D[y[t],{t,4}]+y[t]==g[t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to e^{-\frac {t}{\sqrt {2}}} \left (e^{\sqrt {2} t} \cos \left (\frac {t}{\sqrt {2}}\right ) \int _1^t-\frac {e^{-\frac {K[1]}{\sqrt {2}}} g(K[1]) \left (\cos \left (\frac {K[1]}{\sqrt {2}}\right )+\sin \left (\frac {K[1]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[1]+\cos \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{\frac {K[2]}{\sqrt {2}}} g(K[2]) \left (\cos \left (\frac {K[2]}{\sqrt {2}}\right )-\sin \left (\frac {K[2]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[2]+\sin \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{\frac {K[3]}{\sqrt {2}}} g(K[3]) \left (\cos \left (\frac {K[3]}{\sqrt {2}}\right )+\sin \left (\frac {K[3]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[3]+e^{\sqrt {2} t} \sin \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{-\frac {K[4]}{\sqrt {2}}} g(K[4]) \left (\cos \left (\frac {K[4]}{\sqrt {2}}\right )-\sin \left (\frac {K[4]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[4]+c_1 e^{\sqrt {2} t} \cos \left (\frac {t}{\sqrt {2}}\right )+c_2 \cos \left (\frac {t}{\sqrt {2}}\right )+c_3 \sin \left (\frac {t}{\sqrt {2}}\right )+c_4 e^{\sqrt {2} t} \sin \left (\frac {t}{\sqrt {2}}\right )\right )
\]
✓ Sympy. Time used: 6.745 (sec). Leaf size: 192
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)
\[
y{\left (t \right )} = \left (\left (C_{1} - \frac {\int g{\left (t \right )} e^{- \frac {\sqrt {2} t}{2}} \sin {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \cos {\left (\frac {\sqrt {2} t}{2} \right )} + \left (C_{2} + \frac {\int g{\left (t \right )} e^{- \frac {\sqrt {2} t}{2}} \cos {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \sin {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{\frac {\sqrt {2} t}{2}} + \left (\left (C_{3} + \frac {\int g{\left (t \right )} e^{\frac {\sqrt {2} t}{2}} \sin {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \sin {\left (\frac {\sqrt {2} t}{2} \right )} + \left (C_{4} + \frac {\int g{\left (t \right )} e^{\frac {\sqrt {2} t}{2}} \cos {\left (\frac {2 \sqrt {2} t + \pi }{4} \right )}\, dt}{2}\right ) \cos {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{- \frac {\sqrt {2} t}{2}}
\]