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
:
Monday, January 27, 2025 at 06:12:17 AM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }+y&=g \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 257
dsolve(diff(y(t),t$4)+y(t)=g(t),y(t), singsol=all)
\[
y = \frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+\frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{4}-\frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+\frac {\sqrt {2}\, \left (\int \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )-\sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} g \left (t \right )d t \right ) {\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )}{4}+c_1 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )+c_2 \,{\mathrm e}^{-\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \cos \left (\frac {\sqrt {2}\, t}{2}\right )+c_4 \,{\mathrm e}^{\frac {\sqrt {2}\, t}{2}} \sin \left (\frac {\sqrt {2}\, t}{2}\right )
\]
✓ Solution by Mathematica
Time used: 0.324 (sec). Leaf size: 331
DSolve[D[y[t],{t,4}]+y[t]==g[t],y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to e^{-\frac {t}{\sqrt {2}}} \left (e^{\sqrt {2} t} \cos \left (\frac {t}{\sqrt {2}}\right ) \int _1^t-\frac {e^{-\frac {K[1]}{\sqrt {2}}} g(K[1]) \left (\cos \left (\frac {K[1]}{\sqrt {2}}\right )+\sin \left (\frac {K[1]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[1]+\cos \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{\frac {K[2]}{\sqrt {2}}} g(K[2]) \left (\cos \left (\frac {K[2]}{\sqrt {2}}\right )-\sin \left (\frac {K[2]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[2]+\sin \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{\frac {K[3]}{\sqrt {2}}} g(K[3]) \left (\cos \left (\frac {K[3]}{\sqrt {2}}\right )+\sin \left (\frac {K[3]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[3]+e^{\sqrt {2} t} \sin \left (\frac {t}{\sqrt {2}}\right ) \int _1^t\frac {e^{-\frac {K[4]}{\sqrt {2}}} g(K[4]) \left (\cos \left (\frac {K[4]}{\sqrt {2}}\right )-\sin \left (\frac {K[4]}{\sqrt {2}}\right )\right )}{2 \sqrt {2}}dK[4]+c_1 e^{\sqrt {2} t} \cos \left (\frac {t}{\sqrt {2}}\right )+c_2 \cos \left (\frac {t}{\sqrt {2}}\right )+c_3 \sin \left (\frac {t}{\sqrt {2}}\right )+c_4 e^{\sqrt {2} t} \sin \left (\frac {t}{\sqrt {2}}\right )\right )
\]