Internal problem ID [12819]
Internal file name [OUTPUT/11472_Saturday_November_04_2023_08_47_31_AM_24458766/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 5. The Laplace Transform Method. Exercises 5.5, page 273
Problem number: 7.
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]]
\[ \boxed {y^{\prime \prime }+a^{2} y=\delta \left (x -\pi \right ) f \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(x)y^{\prime } + q(x) y &= F \end {align*}
Where here \begin {align*} p(x) &=0\\ q(x) &=a^{2}\\ F &=f \left (\pi \right ) \delta \left (x -\pi \right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime \prime }+a^{2} y = f \left (\pi \right ) \delta \left (x -\pi \right ) \end {align*}
The domain of \(p(x)=0\) is \[
\{-\infty Solving using the Laplace transform method. Let \begin {align*} \mathcal {L}\left (y\right ) =Y(s) \end {align*}
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 )\\ \mathcal {L}\left (y^{\prime \prime }\right ) &= s^2 Y(s) - y'(0) - s y \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 )-y^{\prime }\left (0\right )-s y \left (0\right )+a^{2} Y \left (s \right ) = f \left (\pi \right ) {\mathrm e}^{-s \pi }\tag {1} \end {align*}
But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y'(0) &=0 \end {align*}
Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+a^{2} Y \left (s \right ) = f \left (\pi \right ) {\mathrm e}^{-s \pi } \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {f \left (\pi \right ) {\mathrm e}^{-s \pi }}{a^{2}+s^{2}} \end {align*}
Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (\frac {f \left (\pi \right ) {\mathrm e}^{-s \pi }}{a^{2}+s^{2}}\right )\\ &= \frac {\operatorname {Heaviside}\left (x -\pi \right ) f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} \end {align*}
Converting the above solution to piecewise it becomes \[
y = \left \{\begin {array}{cc} 0 & x <\pi \\ \frac {f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} & \pi \le x \end {array}\right .
\] Simplifying the solution gives \[
y = \left \{\begin {array}{cc} 0 & x <\pi \\ \frac {f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} & \pi \le x \end {array}\right .
\]
The solution(s) found are the following \begin{align*}
\tag{1} y &= \left \{\begin {array}{cc} 0 & x <\pi \\ \frac {f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} & \pi \le x \end {array}\right . \\
\end{align*} Verification of solutions
\[
y = \left \{\begin {array}{cc} 0 & x <\pi \\ \frac {f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} & \pi \le x \end {array}\right .
\] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+a^{2} y=f \left (\pi \right ) \mathit {Dirac}\left (x -\pi \right ), y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & a^{2}+r^{2}=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {0\pm \left (\sqrt {-4 a^{2}}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (\sqrt {-a^{2}}, -\sqrt {-a^{2}}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{\sqrt {-a^{2}}\, x} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{-\sqrt {-a^{2}}\, x} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right )+y_{p}\left (x \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{\sqrt {-a^{2}}\, x}+c_{2} {\mathrm e}^{-\sqrt {-a^{2}}\, x}+y_{p}\left (x \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (x \right )\hspace {3pt}\textrm {of the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Use variation of parameters to find}\hspace {3pt} y_{p}\hspace {3pt}\textrm {here}\hspace {3pt} f \left (x \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (x \right )=-y_{1}\left (x \right ) \left (\int \frac {y_{2}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right )+y_{2}\left (x \right ) \left (\int \frac {y_{1}\left (x \right ) f \left (x \right )}{W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )}d x \right ), f \left (x \right )=f \left (\pi \right ) \mathit {Dirac}\left (x -\pi \right )\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{\sqrt {-a^{2}}\, x} & {\mathrm e}^{-\sqrt {-a^{2}}\, x} \\ \sqrt {-a^{2}}\, {\mathrm e}^{\sqrt {-a^{2}}\, x} & -\sqrt {-a^{2}}\, {\mathrm e}^{-\sqrt {-a^{2}}\, x} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=-2 \sqrt {-a^{2}} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=\frac {f \left (\pi \right ) \left (\int \mathit {Dirac}\left (x -\pi \right )d x \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=\frac {f \left (\pi \right ) \mathit {Heaviside}\left (x -\pi \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{\sqrt {-a^{2}}\, x}+c_{2} {\mathrm e}^{-\sqrt {-a^{2}}\, x}+\frac {f \left (\pi \right ) \mathit {Heaviside}\left (x -\pi \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{\sqrt {-a^{2}}\, x}+c_{2} {\mathrm e}^{-\sqrt {-a^{2}}\, x}+\frac {f \left (\pi \right ) \mathit {Heaviside}\left (x -\pi \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} +c_{2} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=c_{1} \sqrt {-a^{2}}\, {\mathrm e}^{\sqrt {-a^{2}}\, x}-c_{2} \sqrt {-a^{2}}\, {\mathrm e}^{-\sqrt {-a^{2}}\, x}+\frac {f \left (\pi \right ) \mathit {Dirac}\left (x -\pi \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}+\frac {f \left (\pi \right ) \mathit {Heaviside}\left (x -\pi \right ) \left ({\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}} \sqrt {-a^{2}}+\sqrt {-a^{2}}\, {\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=c_{1} \sqrt {-a^{2}}-c_{2} \sqrt {-a^{2}} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =0, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {f \left (\pi \right ) \mathit {Heaviside}\left (x -\pi \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {f \left (\pi \right ) \mathit {Heaviside}\left (x -\pi \right ) \left (-{\mathrm e}^{\left (\pi -x \right ) \sqrt {-a^{2}}}+{\mathrm e}^{\left (x -\pi \right ) \sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}} \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 6.188 (sec). Leaf size: 24
\[
y \left (x \right ) = \frac {\operatorname {Heaviside}\left (x -\pi \right ) \sin \left (a \left (x -\pi \right )\right ) f \left (\pi \right )}{a}
\]
✓ Solution by Mathematica
Time used: 0.398 (sec). Leaf size: 26
\[
y(x)\to -\frac {f(\pi ) \theta (x-\pi ) \sin (a (\pi -x))}{a}
\]
16.7.2 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(y(x),x$2)+a^2*y(x)=Dirac(x-Pi)*f(x),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
DSolve[{y''[x]+a^2*y[x]==DiracDelta[x-Pi]*f[x],{y[0]==0,y'[0]==0}},y[x],x,IncludeSingularSolutions -> True]