problem 12

Internal problem ID [12795]
Internal ﬁle name [OUTPUT/11448_Saturday_November_04_2023_08_47_22_AM_84888398/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: 12.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second_order_laplace", "second_order_linear_constant_coeﬀ"

\[ \boxed {y^{\prime \prime }-y^{\prime }+6 y=-2 \sin \left (3 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*}

13.12.1 Existence and uniqueness analysis

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) &=-1\\ q(x) &=6\\ F &=-2 \sin \left (3 x \right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }-y^{\prime }+6 y = -2 \sin \left (3 x \right ) \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 )-s Y \left (s \right )+y \left (0\right )+6 Y \left (s \right ) = -\frac {6}{s^{2}+9}\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=0\\ y'(0) &=-1 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+1-s Y \left (s \right )+6 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 {s^{2}+15}{\left (s^{2}+9\right ) \left (s^{2}-s +6\right )} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= \frac {\frac {1}{6}+\frac {13 i \sqrt {23}}{138}}{s -\frac {1}{2}-\frac {i \sqrt {23}}{2}}+\frac {\frac {1}{6}-\frac {13 i \sqrt {23}}{138}}{s -\frac {1}{2}+\frac {i \sqrt {23}}{2}}+\frac {-\frac {1}{6}-\frac {i}{6}}{s -3 i}+\frac {-\frac {1}{6}+\frac {i}{6}}{s +3 i} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {\frac {1}{6}+\frac {13 i \sqrt {23}}{138}}{s -\frac {1}{2}-\frac {i \sqrt {23}}{2}}\right ) &= \frac {\left (13 i \sqrt {23}+23\right ) {\mathrm e}^{\frac {\left (1+i \sqrt {23}\right ) x}{2}}}{138}\\ \mathcal {L}^{-1}\left (\frac {\frac {1}{6}-\frac {13 i \sqrt {23}}{138}}{s -\frac {1}{2}+\frac {i \sqrt {23}}{2}}\right ) &= \frac {\left (23-13 i \sqrt {23}\right ) {\mathrm e}^{\frac {\left (1-i \sqrt {23}\right ) x}{2}}}{138}\\ \mathcal {L}^{-1}\left (\frac {-\frac {1}{6}-\frac {i}{6}}{s -3 i}\right ) &= \left (-\frac {1}{6}-\frac {i}{6}\right ) {\mathrm e}^{3 i x}\\ \mathcal {L}^{-1}\left (\frac {-\frac {1}{6}+\frac {i}{6}}{s +3 i}\right ) &= \left (-\frac {1}{6}+\frac {i}{6}\right ) {\mathrm e}^{-3 i x} \end {align*}

Adding the above results and simplifying gives \[ y=-\frac {\cos \left (3 x \right )}{3}+\frac {\sin \left (3 x \right )}{3}+\frac {{\mathrm e}^{\frac {x}{2}} \left (-13 \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, x}{2}\right )+23 \cos \left (\frac {\sqrt {23}\, x}{2}\right )\right )}{69} \] Simplifying the solution gives \[ y = -\frac {13 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {13 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \\ \end{align*}

Veriﬁcation of solutions

\[ y = -\frac {13 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \] Veriﬁed OK.

13.12.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }-y^{\prime }+6 y=-2 \sin \left (3 x \right ), y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-1\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}-r +6=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {1\pm \left (\sqrt {-23}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {23}}{2}, \frac {1}{2}+\frac {\mathrm {I} \sqrt {23}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right ) \\ \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}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )+c_{2} {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )+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}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right ) & {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right ) \\ \frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{2}-\frac {\sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{2} & \frac {{\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{2}+\frac {{\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{2} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\frac {\sqrt {23}\, {\mathrm e}^{x}}{2} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-\frac {4 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \left (\sin \left (\frac {\sqrt {23}\, x}{2}\right ) \left (\int {\mathrm e}^{-\frac {x}{2}} \sin \left (3 x \right ) \cos \left (\frac {\sqrt {23}\, x}{2}\right )d x \right )-\cos \left (\frac {\sqrt {23}\, x}{2}\right ) \left (\int {\mathrm e}^{-\frac {x}{2}} \sin \left (3 x \right ) \sin \left (\frac {\sqrt {23}\, x}{2}\right )d x \right )\right )}{23} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=-\frac {\cos \left (3 x \right )}{3}+\frac {\sin \left (3 x \right )}{3} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )+c_{2} {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )-\frac {\cos \left (3 x \right )}{3}+\frac {\sin \left (3 x \right )}{3} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )+c_{2} {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )-\frac {\cos \left (3 x \right )}{3}+\frac {\sin \left (3 x \right )}{3} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} -\frac {1}{3} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {c_{1} {\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{2}-\frac {c_{1} {\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{2}+\sin \left (3 x \right )+\cos \left (3 x \right ) \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=-1 \\ {} & {} & -1=\frac {c_{1}}{2}+1+\frac {c_{2} \sqrt {23}}{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =\frac {1}{3}, c_{2} =-\frac {13 \sqrt {23}}{69}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {13 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {13 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \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`

\[ y \left (x \right ) = -\frac {13 \,{\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \]

\[ y(x)\to \frac {1}{69} \left (23 \sin (3 x)-13 \sqrt {23} e^{x/2} \sin \left (\frac {\sqrt {23} x}{2}\right )-23 \cos (3 x)+23 e^{x/2} \cos \left (\frac {\sqrt {23} x}{2}\right )\right ) \]

13.12 problem 12

13.12.1 Existence and uniqueness analysis

13.12.2 Maple step by step solution