4.6 problem Problem 2(f)

Internal problem ID [12313]
Internal file name [OUTPUT/10966_Monday_October_02_2023_02_47_38_AM_95537058/index.tex]

Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section: Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number: Problem 2(f).
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, _with_linear_symmetries]]

\[ \boxed {2 y^{\prime \prime }-3 y^{\prime }+17 y=17 t -1} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*}

4.6.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(t)y^{\prime } + q(t) y &= F \end {align*}

Where here \begin {align*} p(t) &=-{\frac {3}{2}}\\ q(t) &={\frac {17}{2}}\\ F &=\frac {17 t}{2}-\frac {1}{2} \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }-\frac {3 y^{\prime }}{2}+\frac {17 y}{2} = \frac {17 t}{2}-\frac {1}{2} \end {align*}

The domain of \(p(t)=-{\frac {3}{2}}\) 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*} 2 s^{2} Y \left (s \right )-2 y^{\prime }\left (0\right )-2 s y \left (0\right )-3 s Y \left (s \right )+3 y \left (0\right )+17 Y \left (s \right ) = \frac {17}{s^{2}}-\frac {1}{s}\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=-1\\ y'(0) &=2 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} 2 s^{2} Y \left (s \right )-7+2 s -3 s Y \left (s \right )+17 Y \left (s \right ) = \frac {17}{s^{2}}-\frac {1}{s} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = -\frac {2 s^{3}-7 s^{2}+s -17}{s^{2} \left (2 s^{2}-3 s +17\right )} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= \frac {-\frac {19}{34}-\frac {125 i \sqrt {127}}{4318}}{s -\frac {3}{4}-\frac {i \sqrt {127}}{4}}+\frac {-\frac {19}{34}+\frac {125 i \sqrt {127}}{4318}}{s -\frac {3}{4}+\frac {i \sqrt {127}}{4}}+\frac {1}{s^{2}}+\frac {2}{17 s} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {-\frac {19}{34}-\frac {125 i \sqrt {127}}{4318}}{s -\frac {3}{4}-\frac {i \sqrt {127}}{4}}\right ) &= -\frac {\left (125 i \sqrt {127}+2413\right ) {\mathrm e}^{\frac {\left (3+i \sqrt {127}\right ) \left (\frac {3 t}{4 \left (\frac {3}{4}+\frac {i \sqrt {127}}{4}\right )}+\frac {i \sqrt {127}\, t}{3+i \sqrt {127}}\right )}{4}}}{4318}\\ \mathcal {L}^{-1}\left (\frac {-\frac {19}{34}+\frac {125 i \sqrt {127}}{4318}}{s -\frac {3}{4}+\frac {i \sqrt {127}}{4}}\right ) &= -\frac {\left (2413-125 i \sqrt {127}\right ) {\mathrm e}^{\frac {\left (3-i \sqrt {127}\right ) \left (\frac {3 t}{4 \left (\frac {3}{4}-\frac {i \sqrt {127}}{4}\right )}-\frac {i \sqrt {127}\, t}{4 \left (\frac {3}{4}-\frac {i \sqrt {127}}{4}\right )}\right )}{4}}}{4318}\\ \mathcal {L}^{-1}\left (\frac {1}{s^{2}}\right ) &= t\\ \mathcal {L}^{-1}\left (\frac {2}{17 s}\right ) &= {\frac {2}{17}} \end {align*}

Adding the above results and simplifying gives \[ y=\frac {2}{17}+t +\frac {\left (125 \sqrt {127}\, \sin \left (\frac {\sqrt {127}\, t}{4}\right )-2413 \cos \left (\frac {\sqrt {127}\, t}{4}\right )\right ) {\mathrm e}^{\frac {3 t}{4}}}{2159} \] Simplifying the solution gives \[ y = \frac {125 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {125 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {125 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \] Verified OK.

4.6.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [2 y^{\prime \prime }-3 y^{\prime }+17 y=17 t -1, y \left (0\right )=-1, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=2\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=\frac {3 y^{\prime }}{2}-\frac {17 y}{2}+\frac {17 t}{2}-\frac {1}{2} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime }-\frac {3 y^{\prime }}{2}+\frac {17 y}{2}=\frac {17 t}{2}-\frac {1}{2} \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}-\frac {3}{2} r +\frac {17}{2}=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (\frac {3}{2}\right )\pm \left (\sqrt {-\frac {127}{4}}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (\frac {3}{4}-\frac {\mathrm {I} \sqrt {127}}{4}, \frac {3}{4}+\frac {\mathrm {I} \sqrt {127}}{4}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right )+y_{p}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )+c_{2} {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )+y_{p}\left (t \right ) \\ \square & {} & \textrm {Find a particular solution}\hspace {3pt} y_{p}\left (t \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 (t \right )\hspace {3pt}\textrm {is the forcing function}\hspace {3pt} \\ {} & {} & \left [y_{p}\left (t \right )=-y_{1}\left (t \right ) \left (\int \frac {y_{2}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right )+y_{2}\left (t \right ) \left (\int \frac {y_{1}\left (t \right ) f \left (t \right )}{W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )}d t \right ), f \left (t \right )=\frac {17 t}{2}-\frac {1}{2}\right ] \\ {} & \circ & \textrm {Wronskian of solutions of the homogeneous equation}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\left [\begin {array}{cc} {\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right ) & {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \\ \frac {3 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{4}-\frac {{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{4} & \frac {3 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )}{4}+\frac {{\mathrm e}^{\frac {3 t}{4}} \sqrt {127}\, \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{4} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=\frac {\sqrt {127}\, {\mathrm e}^{\frac {3 t}{2}}}{4} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=\frac {2 \sqrt {127}\, {\mathrm e}^{\frac {3 t}{4}} \left (-\cos \left (\frac {\sqrt {127}\, t}{4}\right ) \left (\int \left (17 t -1\right ) {\mathrm e}^{-\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )d t \right )+\sin \left (\frac {\sqrt {127}\, t}{4}\right ) \left (\int \left (17 t -1\right ) {\mathrm e}^{-\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )d t \right )\right )}{127} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=t +\frac {2}{17} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )+c_{2} {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )+t +\frac {2}{17} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )+c_{2} {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )+t +\frac {2}{17} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=-1 \\ {} & {} & -1=c_{1} +\frac {2}{17} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 c_{1} {\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{4}-\frac {c_{1} {\mathrm e}^{\frac {3 t}{4}} \sqrt {127}\, \sin \left (\frac {\sqrt {127}\, t}{4}\right )}{4}+\frac {3 c_{2} {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )}{4}+\frac {c_{2} {\mathrm e}^{\frac {3 t}{4}} \sqrt {127}\, \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{4}+1 \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=2 \\ {} & {} & 2=\frac {3 c_{1}}{4}+1+\frac {c_{2} \sqrt {127}}{4} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {19}{17}, c_{2} =\frac {125 \sqrt {127}}{2159}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\frac {125 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\frac {125 \,{\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right ) \sqrt {127}}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \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: 5.359 (sec). Leaf size: 35

dsolve([2*diff(y(t),t$2)-3*diff(y(t),t)+17*y(t)=17*t-1,y(0) = -1, D(y)(0) = 2],y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {125 \sqrt {127}\, {\mathrm e}^{\frac {3 t}{4}} \sin \left (\frac {\sqrt {127}\, t}{4}\right )}{2159}-\frac {19 \,{\mathrm e}^{\frac {3 t}{4}} \cos \left (\frac {\sqrt {127}\, t}{4}\right )}{17}+t +\frac {2}{17} \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 59

DSolve[{2*y''[t]-3*y'[t]+17*y[t]==17*t-1,{y[0]==-1,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to t+\frac {125 e^{3 t/4} \sin \left (\frac {\sqrt {127} t}{4}\right )}{17 \sqrt {127}}-\frac {19}{17} e^{3 t/4} \cos \left (\frac {\sqrt {127} t}{4}\right )+\frac {2}{17} \]