problem 6(e)

Internal problem ID [11511]
Internal ﬁle name [OUTPUT/10494_Thursday_May_18_2023_04_21_05_AM_22319864/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(e).
ODE order: 2.
ODE degree: 1.

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

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

15.5.1 Existence and uniqueness analysis

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) &=-2\\ q(t) &=2\\ F &={\mathrm e}^{-t} \end {align*}

Hence the ode is \begin {align*} x^{\prime \prime }-2 x^{\prime }+2 x = {\mathrm e}^{-t} \end {align*}

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 )-2 s Y \left (s \right )+2 x \left (0\right )+2 Y \left (s \right ) = \frac {1}{s +1}\tag {1} \end {align*}

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

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

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {2+s}{\left (s +1\right ) \left (s^{2}-2 s +2\right )} \end {align*}

Applying partial fractions decomposition results in \[ Y(s)= \frac {-\frac {1}{10}-\frac {7 i}{10}}{s -1-i}+\frac {-\frac {1}{10}+\frac {7 i}{10}}{s -1+i}+\frac {1}{5 s +5} \] The inverse Laplace of each term above is now found, which gives \begin {align*} \mathcal {L}^{-1}\left (\frac {-\frac {1}{10}-\frac {7 i}{10}}{s -1-i}\right ) &= \left (-\frac {1}{10}-\frac {7 i}{10}\right ) {\mathrm e}^{\left (1+i\right ) t}\\ \mathcal {L}^{-1}\left (\frac {-\frac {1}{10}+\frac {7 i}{10}}{s -1+i}\right ) &= \left (-\frac {1}{10}+\frac {7 i}{10}\right ) {\mathrm e}^{\left (1-i\right ) t}\\ \mathcal {L}^{-1}\left (\frac {1}{5 s +5}\right ) &= \frac {{\mathrm e}^{-t}}{5} \end {align*}

Adding the above results and simplifying gives \[ x=\frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \] Simplifying the solution gives \[ x = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \\ \end{align*}

Veriﬁcation of solutions

\[ x = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \] Veriﬁed OK.

15.5.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime \prime }-2 x^{\prime }+2 x={\mathrm e}^{-t}, x \left (0\right )=0, x^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=1\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}-2 r +2=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {2\pm \left (\sqrt {-4}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (1-\mathrm {I}, 1+\mathrm {I}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x_{1}\left (t \right )={\mathrm e}^{t} \cos \left (t \right ) \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & x_{2}\left (t \right )={\mathrm e}^{t} \sin \left (t \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}^{t} \cos \left (t \right )+c_{2} {\mathrm e}^{t} \sin \left (t \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 )={\mathrm e}^{-t}\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}^{t} \cos \left (t \right ) & {\mathrm e}^{t} \sin \left (t \right ) \\ {\mathrm e}^{t} \cos \left (t \right )-{\mathrm e}^{t} \sin \left (t \right ) & {\mathrm e}^{t} \sin \left (t \right )+{\mathrm e}^{t} \cos \left (t \right ) \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (x_{1}\left (t \right ), x_{2}\left (t \right )\right )={\mathrm e}^{2 t} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} x_{p}\left (t \right ) \\ {} & {} & x_{p}\left (t \right )=-{\mathrm e}^{t} \left (\cos \left (t \right ) \left (\int \sin \left (t \right ) {\mathrm e}^{-2 t}d t \right )-\sin \left (t \right ) \left (\int \cos \left (t \right ) {\mathrm e}^{-2 t}d t \right )\right ) \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & x_{p}\left (t \right )=\frac {{\mathrm e}^{-t}}{5} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & x=c_{1} {\mathrm e}^{t} \cos \left (t \right )+c_{2} {\mathrm e}^{t} \sin \left (t \right )+\frac {{\mathrm e}^{-t}}{5} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} x=c_{1} {\mathrm e}^{t} \cos \left (t \right )+c_{2} {\mathrm e}^{t} \sin \left (t \right )+\frac {{\mathrm e}^{-t}}{5} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} x \left (0\right )=0 \\ {} & {} & 0=c_{1} +\frac {1}{5} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & x^{\prime }=c_{1} {\mathrm e}^{t} \cos \left (t \right )-c_{1} {\mathrm e}^{t} \sin \left (t \right )+c_{2} {\mathrm e}^{t} \sin \left (t \right )+c_{2} {\mathrm e}^{t} \cos \left (t \right )-\frac {{\mathrm e}^{-t}}{5} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} x^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=1 \\ {} & {} & 1=c_{1} -\frac {1}{5}+c_{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {1}{5}, c_{2} =\frac {7}{5}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & x=\frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x=\frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \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`

\[ x \left (t \right ) = \frac {{\mathrm e}^{-t}}{5}+\frac {\left (-\cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{t}}{5} \]

15.5 problem 6(e)

15.5.1 Existence and uniqueness analysis

15.5.2 Maple step by step solution