14.18.7 problem 7
Internal
problem
ID
[2691]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
2.
Second
order
differential
equations.
Section
2.11,
Differential
equations
with
discontinuous
right-hand
sides.
Excercises
page
243
Problem
number
:
7
Date
solved
:
Tuesday, January 28, 2025 at 02:36:26 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+y^{\prime }+7 y&=\left \{\begin {array}{cc} t & 0\le t <2 \\ 0 & 2\le t \end {array}\right . \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 1.734 (sec). Leaf size: 146
dsolve([diff(y(t),t$2)+diff(y(t),t)+7*y(t)=piecewise(0<=t and t<2,t,t>=2,0),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[
y = -\frac {\left (\left \{\begin {array}{cc} 13 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )-9 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )-63 t +9 & t <2 \\ \left (13 \sqrt {3}\, \sin \left (3 \sqrt {3}\right )-234 \,{\mathrm e}-9 \cos \left (3 \sqrt {3}\right )\right ) {\mathrm e}^{-1} & t =2 \\ -27 \sqrt {3}\, {\mathrm e}^{-\frac {t}{2}+1} \sin \left (\frac {3 \sqrt {3}\, \left (t -2\right )}{2}\right )+13 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )-117 \,{\mathrm e}^{-\frac {t}{2}+1} \cos \left (\frac {3 \sqrt {3}\, \left (t -2\right )}{2}\right )-9 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right ) & 2<t \end {array}\right .\right )}{441}
\]
✓ Solution by Mathematica
Time used: 0.088 (sec). Leaf size: 153
DSolve[{D[y[t],{t,2}]+D[y[t],t]+7*y[t]==Piecewise[{{t,0<=t<2},{0,t>=2}}],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{441} e^{-t/2} \left (9 e^{t/2} (7 t-1)+9 \cos \left (\frac {3 \sqrt {3} t}{2}\right )-13 \sqrt {3} \sin \left (\frac {3 \sqrt {3} t}{2}\right )\right ) & 0<t\leq 2 \\ \frac {1}{441} e^{-t/2} \left (117 e \cos \left (\frac {3}{2} \sqrt {3} (t-2)\right )+9 \cos \left (\frac {3 \sqrt {3} t}{2}\right )+\sqrt {3} \left (27 e \sin \left (\frac {3}{2} \sqrt {3} (t-2)\right )-13 \sin \left (\frac {3 \sqrt {3} t}{2}\right )\right )\right ) & t>2 \\ \end {array} \\ \end {array}
\]