Internal problem ID [12787]
Internal file name [OUTPUT/11440_Saturday_November_04_2023_08_47_20_AM_85677221/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number: 4.
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]]
\[ \boxed {y^{\prime \prime }-9 y=2 \sin \left (3 x \right )} \] Since no initial conditions are explicitly given, then let \begin {align*} y \left (0\right )&= c_{1} \\ y'(0) &= c_{2} \end {align*}
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 )-9 Y \left (s \right ) = \frac {6}{s^{2}+9}\tag {1} \end {align*}
But the initial conditions are \begin {align*} y \left (0\right )&=c_{1}\\ y'(0) &=c_{2} \end {align*}
Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )-c_{2} -s c_{1} -9 Y \left (s \right ) = \frac {6}{s^{2}+9} \end {align*}
Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {c_{1} s^{3}+c_{2} s^{2}+9 s c_{1} +9 c_{2} +6}{\left (s^{2}+9\right ) \left (s^{2}-9\right )} \end {align*}
Applying partial fractions decomposition results in \[ Y(s)= \frac {\frac {c_{1}}{2}+\frac {c_{2}}{6}+\frac {1}{18}}{s -3}+\frac {i}{18 s -54 i}-\frac {i}{18 \left (s +3 i\right )}+\frac {\frac {c_{1}}{2}-\frac {c_{2}}{6}-\frac {1}{18}}{s +3} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {\frac {c_{1}}{2}+\frac {c_{2}}{6}+\frac {1}{18}}{s -3}\right ) &= \frac {\left (9 c_{1} +3 c_{2} +1\right ) {\mathrm e}^{3 x}}{18}\\ \mathcal {L}^{-1}\left (\frac {i}{18 s -54 i}\right ) &= \frac {i {\mathrm e}^{3 i x}}{18}\\ \mathcal {L}^{-1}\left (-\frac {i}{18 \left (s +3 i\right )}\right ) &= -\frac {i {\mathrm e}^{-3 i x}}{18}\\ \mathcal {L}^{-1}\left (\frac {\frac {c_{1}}{2}-\frac {c_{2}}{6}-\frac {1}{18}}{s +3}\right ) &= \frac {\left (9 c_{1} -3 c_{2} -1\right ) {\mathrm e}^{-3 x}}{18} \end {align*}
Adding the above results and simplifying gives \[ y=-\frac {\sin \left (3 x \right )}{9}+c_{1} \cosh \left (3 x \right )+\frac {\sinh \left (3 x \right ) \left (1+3 c_{2} \right )}{9} \] Simplifying the solution gives \[ y = -\frac {\sin \left (3 x \right )}{9}+c_{1} \cosh \left (3 x \right )+\frac {\sinh \left (3 x \right ) \left (1+3 c_{2} \right )}{9} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\sin \left (3 x \right )}{9}+c_{1} \cosh \left (3 x \right )+\frac {\sinh \left (3 x \right ) \left (1+3 c_{2} \right )}{9} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\sin \left (3 x \right )}{9}+c_{1} \cosh \left (3 x \right )+\frac {\sinh \left (3 x \right ) \left (1+3 c_{2} \right )}{9} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }-9 y=2 \sin \left (3 x \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} \\ {} & {} & r^{2}-9=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \left (r -3\right ) \left (r +3\right )=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-3, 3\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{-3 x} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{3 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}^{-3 x}+c_{2} {\mathrm e}^{3 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 )=2 \sin \left (3 x \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}^{-3 x} & {\mathrm e}^{3 x} \\ -3 \,{\mathrm e}^{-3 x} & 3 \,{\mathrm e}^{3 x} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=6 \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-\frac {{\mathrm e}^{-3 x} \left (\int \sin \left (3 x \right ) {\mathrm e}^{3 x}d x \right )}{3}+\frac {{\mathrm e}^{3 x} \left (\int {\mathrm e}^{-3 x} \sin \left (3 x \right )d x \right )}{3} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=-\frac {\sin \left (3 x \right )}{9} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-3 x}+c_{2} {\mathrm e}^{3 x}-\frac {\sin \left (3 x \right )}{9} \end {array} \]
Maple trace
`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`
✓ Solution by Maple
Time used: 5.781 (sec). Leaf size: 30
dsolve(diff(y(x),x$2)-9*y(x)=2*sin(3*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\sin \left (3 x \right )}{9}+y \left (0\right ) \cosh \left (3 x \right )+\frac {\sinh \left (3 x \right ) \left (1+3 D\left (y \right )\left (0\right )\right )}{9} \]
✓ Solution by Mathematica
Time used: 0.032 (sec). Leaf size: 30
DSolve[y''[x]-9*y[x]==2*Sin[3*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{9} \sin (3 x)+c_1 e^{3 x}+c_2 e^{-3 x} \]