73.18.13 problem 27.1 (m)
Internal
problem
ID
[15518]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
27.
Differentiation
and
the
Laplace
transform.
Additional
Exercises.
page
496
Problem
number
:
27.1
(m)
Date
solved
:
Thursday, March 13, 2025 at 06:10:47 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime \prime }-27 y&={\mathrm e}^{-3 t} \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=3\\ y^{\prime \prime }\left (0\right )&=4 \end{align*}
✓ Maple. Time used: 11.371 (sec). Leaf size: 46
ode:=diff(diff(diff(y(t),t),t),t)-27*y(t) = exp(-3*t);
ic:=y(0) = 2, D(y)(0) = 3, (D@@2)(y)(0) = 4;
dsolve([ode,ic],y(t),method='laplace');
\[
y = \frac {14 \sqrt {3}\, {\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{81}+\frac {70 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{81}+\frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81}
\]
✓ Mathematica. Time used: 0.363 (sec). Leaf size: 376
ode=D[ y[t],{t,3}]-27*y[t]==Exp[-3*t];
ic={y[0]==2,Derivative[1][y][0] ==3,Derivative[2][y][0] ==4};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\[
y(t)\to \frac {1}{162} e^{-3 t} \left (-162 e^{3 t/2} \cos \left (\frac {3 \sqrt {3} t}{2}\right ) \int _1^0-\frac {e^{-\frac {3 K[1]}{2}} \left (\sqrt {3} \cos \left (\frac {3}{2} \sqrt {3} K[1]\right )-3 \sin \left (\frac {3}{2} \sqrt {3} K[1]\right )\right )}{27 \sqrt {3}}dK[1]+162 e^{3 t/2} \cos \left (\frac {3 \sqrt {3} t}{2}\right ) \int _1^t-\frac {e^{-\frac {3 K[1]}{2}} \left (\sqrt {3} \cos \left (\frac {3}{2} \sqrt {3} K[1]\right )-3 \sin \left (\frac {3}{2} \sqrt {3} K[1]\right )\right )}{27 \sqrt {3}}dK[1]-162 e^{3 t/2} \sin \left (\frac {3 \sqrt {3} t}{2}\right ) \int _1^0-\frac {e^{-\frac {3 K[2]}{2}} \left (3 \cos \left (\frac {3}{2} \sqrt {3} K[2]\right )+\sqrt {3} \sin \left (\frac {3}{2} \sqrt {3} K[2]\right )\right )}{27 \sqrt {3}}dK[2]+162 e^{3 t/2} \sin \left (\frac {3 \sqrt {3} t}{2}\right ) \int _1^t-\frac {e^{-\frac {3 K[2]}{2}} \left (3 \cos \left (\frac {3}{2} \sqrt {3} K[2]\right )+\sqrt {3} \sin \left (\frac {3}{2} \sqrt {3} K[2]\right )\right )}{27 \sqrt {3}}dK[2]+187 e^{6 t}+30 \sqrt {3} e^{3 t/2} \sin \left (\frac {3 \sqrt {3} t}{2}\right )+138 e^{3 t/2} \cos \left (\frac {3 \sqrt {3} t}{2}\right )-1\right )
\]
✓ Sympy. Time used: 0.269 (sec). Leaf size: 61
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-27*y(t) + Derivative(y(t), (t, 3)) - exp(-3*t),0)
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 3, Subs(Derivative(y(t), (t, 2)), t, 0): 4}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (\frac {14 \sqrt {3} \sin {\left (\frac {3 \sqrt {3} t}{2} \right )}}{81} + \frac {70 \cos {\left (\frac {3 \sqrt {3} t}{2} \right )}}{81}\right ) e^{- \frac {3 t}{2}} + \frac {187 e^{3 t}}{162} - \frac {e^{- 3 t}}{54}
\]