6.1 problem 21
Internal
problem
ID
[7318]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
7
THE
LAPLACE
TRANSFORM.
7.3.1
TRANSLATION
ON
THE
s-AXIS.
Page
297
Problem
number
:
21
Date
solved
:
Friday, October 11, 2024 at 08:07:44 AM
CAS
classification
:
[[_linear, `class A`]]
Solve
\begin{align*} y^{\prime }+4 y&={\mathrm e}^{-4 t} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=2 \end{align*}
Solving using the Laplace transform method. Let
\[ \mathcal {L}\left (y\right ) =Y(s) \]
Taking the Laplace transform of the ode
and using the relations that
\begin{align*} \mathcal {L}\left (y^{\prime }\right )&= s Y(s) - y \left (0\right ) \end{align*}
The given ode now becomes an algebraic equation in the Laplace domain
\begin{align*} s Y \left (s \right )-y \left (0\right )+4 Y \left (s \right ) = \frac {1}{s +4}\tag {1} \end{align*}
Replacing initial condition gives
\begin{align*} s Y \left (s \right )-2+4 Y \left (s \right ) = \frac {1}{s +4} \end{align*}
Solving for \(Y(s)\) gives
\begin{align*} Y(s) = \frac {9+2 s}{\left (s +4\right )^{2}} \end{align*}
Applying partial fractions decomposition results in
\[ Y(s)= \frac {1}{\left (s +4\right )^{2}}+\frac {2}{s +4} \]
The inverse Laplace of each term above
is now found, which gives
\begin{align*} \mathcal {L}^{-1}\left (\frac {1}{\left (s +4\right )^{2}}\right ) &= t \,{\mathrm e}^{-4 t}\\ \mathcal {L}^{-1}\left (\frac {2}{s +4}\right ) &= 2 \,{\mathrm e}^{-4 t}\end{align*}
Adding the above results and simplifying gives
\[ y=\left (t +2\right ) {\mathrm e}^{-4 t} \]
6.1.1 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y \left (t \right )+4 y \left (t \right )={\mathrm e}^{-4 t}, y \left (0\right )=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}y \left (t \right )=-4 y \left (t \right )+{\mathrm e}^{-4 t} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (t \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & \frac {d}{d t}y \left (t \right )+4 y \left (t \right )={\mathrm e}^{-4 t} \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (t \right ) \\ {} & {} & \mu \left (t \right ) \left (\frac {d}{d t}y \left (t \right )+4 y \left (t \right )\right )=\mu \left (t \right ) {\mathrm e}^{-4 t} \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d t}\left (y \left (t \right ) \mu \left (t \right )\right ) \\ {} & {} & \mu \left (t \right ) \left (\frac {d}{d t}y \left (t \right )+4 y \left (t \right )\right )=\left (\frac {d}{d t}y \left (t \right )\right ) \mu \left (t \right )+y \left (t \right ) \left (\frac {d}{d t}\mu \left (t \right )\right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \frac {d}{d t}\mu \left (t \right ) \\ {} & {} & \frac {d}{d t}\mu \left (t \right )=4 \mu \left (t \right ) \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (t \right )={\mathrm e}^{4 t} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}\left (y \left (t \right ) \mu \left (t \right )\right )\right )d t =\int \mu \left (t \right ) {\mathrm e}^{-4 t}d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \left (t \right ) \mu \left (t \right )=\int \mu \left (t \right ) {\mathrm e}^{-4 t}d t +\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (t \right ) \\ {} & {} & y \left (t \right )=\frac {\int \mu \left (t \right ) {\mathrm e}^{-4 t}d t +\mathit {C1}}{\mu \left (t \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (t \right )={\mathrm e}^{4 t} \\ {} & {} & y \left (t \right )=\frac {\int {\mathrm e}^{4 t} {\mathrm e}^{-4 t}d t +\mathit {C1}}{{\mathrm e}^{4 t}} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y \left (t \right )=\frac {t +\mathit {C1}}{{\mathrm e}^{4 t}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (t \right )={\mathrm e}^{-4 t} \left (t +\mathit {C1} \right ) \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=2 \\ {} & {} & 2=\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \mathit {C1} =2 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_C1} =2\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y \left (t \right )={\mathrm e}^{-4 t} \left (t +2\right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y \left (t \right )={\mathrm e}^{-4 t} \left (t +2\right ) \end {array} \]
6.1.2 Maple trace
` Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful `
6.1.3 Maple dsolve solution
Solving time : 0.102
(sec)
Leaf size : 12
dsolve ([ diff ( y ( t ), t )+4* y ( t ) = exp (-4* t ),
op ([ y (0) = 2])],
y(t),method=laplace)
\[
y = \left (t +2\right ) {\mathrm e}^{-4 t}
\]
6.1.4 Mathematica DSolve solution
Solving time : 0.053
(sec)
Leaf size : 14
DSolve [{ D [ y [ t ], t ]+4* y [ t ]== Exp [-4* t ],{ y [0]==2}},
y[t],t,IncludeSingularSolutions-> True ]
\[
y(t)\to e^{-4 t} (t+2)
\]