Internal problem ID [11513]
Internal file name [OUTPUT/10496_Thursday_May_18_2023_04_21_07_AM_2733894/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page
156
Problem number: 6(g).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_laplace", "second_order_linear_constant_coeff"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x=1-\operatorname {Heaviside}\left (t -5\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} x^{\prime \prime } + p(t)x^{\prime } + q(t) x &= F \end {align*}
Where here \begin {align*} p(t) &={\frac {2}{5}}\\ q(t) &=2\\ F &=1-\operatorname {Heaviside}\left (t -5\right ) \end {align*}
Hence the ode is \begin {align*} x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x = 1-\operatorname {Heaviside}\left (t -5\right ) \end {align*}
The domain of \(p(t)={\frac {2}{5}}\) is \[
\{-\infty Solving using the Laplace transform method. Let \begin {align*} \mathcal {L}\left (x\right ) =Y(s) \end {align*}
Taking the Laplace transform of the ode and using the relations that \begin {align*} \mathcal {L}\left (x^{\prime }\right ) &= s Y(s) - x \left (0\right )\\ \mathcal {L}\left (x^{\prime \prime }\right ) &= s^2 Y(s) - x'(0) - s x \left (0\right ) \end {align*}
The given ode now becomes an algebraic equation in the Laplace domain \begin {align*} s^{2} Y \left (s \right )-x^{\prime }\left (0\right )-s x \left (0\right )+\frac {2 s Y \left (s \right )}{5}-\frac {2 x \left (0\right )}{5}+2 Y \left (s \right ) = \frac {1-{\mathrm e}^{-5 s}}{s}\tag {1} \end {align*}
But the initial conditions are \begin {align*} x \left (0\right )&=0\\ x'(0) &=0 \end {align*}
Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+\frac {2 s Y \left (s \right )}{5}+2 Y \left (s \right ) = \frac {1-{\mathrm e}^{-5 s}}{s} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = -\frac {5 \left (-1+{\mathrm e}^{-5 s}\right )}{s \left (5 s^{2}+2 s +10\right )} \end {align*}
Taking the inverse Laplace transform gives \begin {align*} x&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (-\frac {5 \left (-1+{\mathrm e}^{-5 s}\right )}{s \left (5 s^{2}+2 s +10\right )}\right )\\ &= \frac {1}{2}-\frac {{\mathrm e}^{-\frac {t}{5}} \left (7 \cos \left (\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )\right )}{14}+\left (\frac {1}{100}+\frac {i}{700}\right ) \left (-49+7 i+25 \,{\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (24-7 i\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )}\right ) \operatorname {Heaviside}\left (t -5\right ) \end {align*}
Converting the above solution to piecewise it becomes \[
x = \left \{\begin {array}{cc} \frac {1}{2}-\frac {{\mathrm e}^{-\frac {t}{5}} \left (7 \cos \left (\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )\right )}{14} & t <5 \\ \frac {1}{2}-\frac {{\mathrm e}^{-\frac {t}{5}} \left (7 \cos \left (\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )\right )}{14}+\left (\frac {1}{100}+\frac {i}{700}\right ) \left (-49+7 i+25 \,{\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (24-7 i\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )}\right ) & 5\le t \end {array}\right .
\] Simplifying the solution gives \[
x = \frac {\left (\left \{\begin {array}{cc} 1 & t <5 \\ \left (\frac {1}{2}+\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{2}-\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )} & 5\le t \end {array}\right .\right )}{2}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}
\]
The solution(s) found are the following \begin{align*}
\tag{1} x &= \frac {\left (\left \{\begin {array}{cc} 1 & t <5 \\ \left (\frac {1}{2}+\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{2}-\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )} & 5\le t \end {array}\right .\right )}{2}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14} \\
\end{align*} Verification of solutions
\[
x = \frac {\left (\left \{\begin {array}{cc} 1 & t <5 \\ \left (\frac {1}{2}+\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{2}-\frac {i}{14}\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )} & 5\le t \end {array}\right .\right )}{2}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}
\] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x=1-\mathit {Heaviside}\left (t -5\right ), x \left (0\right )=0, x^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & x^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+\frac {2}{5} r +2=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-\frac {2}{5}\right )\pm \left (\sqrt {-\frac {196}{25}}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {1}{5}-\frac {7 \,\mathrm {I}}{5}, -\frac {1}{5}+\frac {7 \,\mathrm {I}}{5}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x_{1}\left (t \right )={\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x_{2}\left (t \right )={\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & x=c_{1} x_{1}\left (t \right )+c_{2} x_{2}\left (t \right )+x_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x=c_{1} {\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )+c_{2} {\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )+x_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} x_{p}\left (t \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} x_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [x_{p}\left (t \right )=-x_{1}\left (t \right ) \left (\int \frac {x_{2}\left (t \right ) f \left (t \right )}{W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )}d t \right )+x_{2}\left (t \right ) \left (\int \frac {x_{1}\left (t \right ) f \left (t \right )}{W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=1-\mathit {Heaviside}\left (t -5\right )\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right ) & {\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right ) \\ -\frac {{\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )}{5}-\frac {7 \,{\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )}{5} & -\frac {{\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )}{5}+\frac {7 \,{\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )}{5} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )=\frac {7 \,{\mathrm e}^{-\frac {2 t}{5}}}{5} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} x_{p}\left (t \right ) \\ {} & {} & x_{p}\left (t \right )=\frac {5 \,{\mathrm e}^{-\frac {t}{5}} \left (\cos \left (\frac {7 t}{5}\right ) \left (\int \left (-1+\mathit {Heaviside}\left (t -5\right )\right ) \sin \left (\frac {7 t}{5}\right ) {\mathrm e}^{\frac {t}{5}}d t \right )-\sin \left (\frac {7 t}{5}\right ) \left (\int \left (-1+\mathit {Heaviside}\left (t -5\right )\right ) \cos \left (\frac {7 t}{5}\right ) {\mathrm e}^{\frac {t}{5}}d t \right )\right )}{7} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & x_{p}\left (t \right )=\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}-\frac {\mathit {Heaviside}\left (t -5\right )}{2}+\frac {1}{2} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & x=c_{1} {\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )+c_{2} {\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )+\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}-\frac {\mathit {Heaviside}\left (t -5\right )}{2}+\frac {1}{2} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} x=c_{1} {\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )+c_{2} {\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )+\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}-\frac {\mathit {Heaviside}\left (t -5\right )}{2}+\frac {1}{2} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} x \left (0\right )=0 \\ {} & {} & 0=\frac {1}{2}+c_{1} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & x^{\prime }=-\frac {c_{1} {\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )}{5}-\frac {7 c_{1} {\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )}{5}-\frac {c_{2} {\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )}{5}+\frac {7 c_{2} {\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )}{5}+\frac {\left (-\frac {7 \left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \sin \left (\frac {7 t}{5}\right )}{5}+\frac {\cos \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{5}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}+\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Dirac}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}-\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{10}-\frac {\mathit {Dirac}\left (t -5\right )}{2} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} x^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=-\frac {c_{1}}{5}+\frac {7 c_{2}}{5} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {1}{2}, c_{2} =-\frac {1}{14}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & x=-\frac {{\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )}{2}-\frac {{\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )}{14}+\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}-\frac {\mathit {Heaviside}\left (t -5\right )}{2}+\frac {1}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x=-\frac {{\mathrm e}^{-\frac {t}{5}} \cos \left (\frac {7 t}{5}\right )}{2}-\frac {{\mathrm e}^{-\frac {t}{5}} \sin \left (\frac {7 t}{5}\right )}{14}+\frac {\left (\left (\cos \left (7\right )-\frac {\sin \left (7\right )}{7}\right ) \cos \left (\frac {7 t}{5}\right )+\frac {\sin \left (\frac {7 t}{5}\right ) \left (\cos \left (7\right )+7 \sin \left (7\right )\right )}{7}\right ) \mathit {Heaviside}\left (t -5\right ) {\mathrm e}^{1-\frac {t}{5}}}{2}-\frac {\mathit {Heaviside}\left (t -5\right )}{2}+\frac {1}{2} \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 6.672 (sec). Leaf size: 57
\[
x \left (t \right ) = \frac {1}{2}+\left (\frac {1}{4}+\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{4}-\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}-\frac {\operatorname {Heaviside}\left (t -5\right )}{2}
\]
✓ Solution by Mathematica
Time used: 0.07 (sec). Leaf size: 91
\[
x(t)\to -\frac {1}{14} e^{-t/5} \left (-\theta (5-t) \left (7 e^{t/5}+e \sin \left (7-\frac {7 t}{5}\right )-7 e \cos \left (7-\frac {7 t}{5}\right )\right )+e \sin \left (7-\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )-7 e \cos \left (7-\frac {7 t}{5}\right )+7 \cos \left (\frac {7 t}{5}\right )\right )
\]
15.7.2 Maple step by step solution
`Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
trying high order exact linear fully integrable
trying differential order: 2; linear nonhomogeneous with symmetry [0,1]
trying a double symmetry of the form [xi=0, eta=F(x)]
-> Try solving first the homogeneous part of the ODE
checking if the LODE has constant coefficients
<- constant coefficients successful
<- solving first the homogeneous part of the ODE successful`
dsolve([diff(x(t),t$2)+4/10*diff(x(t),t)+2*x(t)=1-Heaviside(t-5),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
DSolve[{x''[t]+4/10*x'[t]+2*x[t]==1-UnitStep[t-5],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]