problem Problem 3(j)

Internal problem ID [12331]
Internal ﬁle name [OUTPUT/10984_Monday_October_02_2023_02_47_43_AM_68847940/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 3(j).
ODE order: 2.
ODE degree: 1.

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

\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (t -3\right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {2}{3}}, y^{\prime }\left (0\right ) = 1\right ] \end {align*}

4.24.1 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) &=4\\ q(t) &=3\\ F &=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (t -3\right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+4 y^{\prime }+3 y = \operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\right )+\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (t -3\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 )+4 s Y \left (s \right )-4 y \left (0\right )+3 Y \left (s \right ) = \frac {1-{\mathrm e}^{-s}+{\mathrm e}^{-2 s}-{\mathrm e}^{-3 s}}{s}\tag {1} \end {align*}

But the initial conditions are \begin {align*} y \left (0\right )&=-{\frac {2}{3}}\\ y'(0) &=1 \end {align*}

Substituting these initial conditions in above in Eq (1) gives \begin {align*} s^{2} Y \left (s \right )+\frac {5}{3}+\frac {2 s}{3}+4 s Y \left (s \right )+3 Y \left (s \right ) = \frac {1-{\mathrm e}^{-s}+{\mathrm e}^{-2 s}-{\mathrm e}^{-3 s}}{s} \end {align*}

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = -\frac {2 s^{2}+3 \,{\mathrm e}^{-s}-3 \,{\mathrm e}^{-2 s}+3 \,{\mathrm e}^{-3 s}+5 s -3}{3 s \left (s^{2}+4 s +3\right )} \end {align*}

Taking the inverse Laplace transform gives \begin {align*} y&= \mathcal {L}^{-1}\left (Y(s)\right )\\ &= \mathcal {L}^{-1}\left (-\frac {2 s^{2}+3 \,{\mathrm e}^{-s}-3 \,{\mathrm e}^{-2 s}+3 \,{\mathrm e}^{-3 s}+5 s -3}{3 s \left (s^{2}+4 s +3\right )}\right )\\ &= \frac {\operatorname {Heaviside}\left (-t +3\right )}{3}-{\mathrm e}^{-t}+\frac {\left (-{\mathrm e}^{-3 t +9}+3 \,{\mathrm e}^{-t +3}\right ) \operatorname {Heaviside}\left (t -3\right )}{6}+\frac {\operatorname {Heaviside}\left (t -2\right ) \left (2+{\mathrm e}^{-3 t +6}-3 \,{\mathrm e}^{-t +2}\right )}{6}+\frac {\left (-2-{\mathrm e}^{-3 t +3}+3 \,{\mathrm e}^{-t +1}\right ) \operatorname {Heaviside}\left (t -1\right )}{6} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} -{\mathrm e}^{-t}+\frac {1}{3} & t <1 \\ -{\mathrm e}^{-t}-\frac {{\mathrm e}^{-3 t +3}}{6}+\frac {{\mathrm e}^{-t +1}}{2} & t <2 \\ -{\mathrm e}^{-t}+\frac {1}{3}+\frac {{\mathrm e}^{-3 t +6}}{6}-\frac {{\mathrm e}^{-t +2}}{2}-\frac {{\mathrm e}^{-3 t +3}}{6}+\frac {{\mathrm e}^{-t +1}}{2} & t <3 \\ -\frac {5 \,{\mathrm e}^{-3}}{6}+\frac {2}{3}-\frac {{\mathrm e}^{-1}}{2}-\frac {{\mathrm e}^{-6}}{6}+\frac {{\mathrm e}^{-2}}{2} & t =3 \\ -{\mathrm e}^{-t}-\frac {{\mathrm e}^{-3 t +9}}{6}+\frac {{\mathrm e}^{-t +3}}{2}+\frac {{\mathrm e}^{-3 t +6}}{6}-\frac {{\mathrm e}^{-t +2}}{2}-\frac {{\mathrm e}^{-3 t +3}}{6}+\frac {{\mathrm e}^{-t +1}}{2} & 3

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\left \{\begin {array}{cc} 3 \,{\mathrm e}^{-t}-1 & t <1 \\ 3 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{-3 t +3}}{2}-\frac {3 \,{\mathrm e}^{-t +1}}{2} & t <2 \\ 3 \,{\mathrm e}^{-t}-1-\frac {{\mathrm e}^{-3 t +6}}{2}+\frac {3 \,{\mathrm e}^{-t +2}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2}-\frac {3 \,{\mathrm e}^{-t +1}}{2} & t <3 \\ \frac {5 \,{\mathrm e}^{-3}}{2}-2+\frac {3 \,{\mathrm e}^{-1}}{2}+\frac {{\mathrm e}^{-6}}{2}-\frac {3 \,{\mathrm e}^{-2}}{2} & t &=3 \\ 3 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{-3 t +9}}{2}-\frac {3 \,{\mathrm e}^{-t +3}}{2}-\frac {{\mathrm e}^{-3 t +6}}{2}+\frac {3 \,{\mathrm e}^{-t +2}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2}-\frac {3 \,{\mathrm e}^{-t +1}}{2} & 3

Veriﬁcation of solutions

\[ y = -\frac {\left (\left \{\begin {array}{cc} 3 \,{\mathrm e}^{-t}-1 & t <1 \\ 3 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{-3 t +3}}{2}-\frac {3 \,{\mathrm e}^{-t +1}}{2} & t <2 \\ 3 \,{\mathrm e}^{-t}-1-\frac {{\mathrm e}^{-3 t +6}}{2}+\frac {3 \,{\mathrm e}^{-t +2}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2}-\frac {3 \,{\mathrm e}^{-t +1}}{2} & t <3 \\ \frac {5 \,{\mathrm e}^{-3}}{2}-2+\frac {3 \,{\mathrm e}^{-1}}{2}+\frac {{\mathrm e}^{-6}}{2}-\frac {3 \,{\mathrm e}^{-2}}{2} & t =3 \\ 3 \,{\mathrm e}^{-t}+\frac {{\mathrm e}^{-3 t +9}}{2}-\frac {3 \,{\mathrm e}^{-t +3}}{2}-\frac {{\mathrm e}^{-3 t +6}}{2}+\frac {3 \,{\mathrm e}^{-t +2}}{2}+\frac {{\mathrm e}^{-3 t +3}}{2}-\frac {3 \,{\mathrm e}^{-t +1}}{2} & 3

4.24.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}y^{\prime }+4 y^{\prime }+3 y=\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -1\right )+\mathit {Heaviside}\left (t -2\right )-\mathit {Heaviside}\left (t -3\right ), y \left (0\right )=-\frac {2}{3}, y^{\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 \\ {} & {} & \frac {d}{d t}y^{\prime } \\ \bullet & {} & \textrm {Characteristic polynomial of homogeneous ODE}\hspace {3pt} \\ {} & {} & r^{2}+4 r +3=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \left (r +3\right ) \left (r +1\right )=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-3, -1\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{-3 t} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{-t} \\ \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}^{-3 t}+c_{2} {\mathrm e}^{-t}+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 )=\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -1\right )+\mathit {Heaviside}\left (t -2\right )-\mathit {Heaviside}\left (t -3\right )\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}^{-3 t} & {\mathrm e}^{-t} \\ -3 \,{\mathrm e}^{-3 t} & -{\mathrm e}^{-t} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (t \right ), y_{2}\left (t \right )\right )=2 \,{\mathrm e}^{-4 t} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (t \right ) \\ {} & {} & y_{p}\left (t \right )=-\frac {{\mathrm e}^{-3 t} \left (\int \left (\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -1\right )+\mathit {Heaviside}\left (t -2\right )-\mathit {Heaviside}\left (t -3\right )\right ) {\mathrm e}^{3 t}d t \right )}{2}+\frac {{\mathrm e}^{-t} \left (\int \left (\mathit {Heaviside}\left (t \right )-\mathit {Heaviside}\left (t -1\right )+\mathit {Heaviside}\left (t -2\right )-\mathit {Heaviside}\left (t -3\right )\right ) {\mathrm e}^{t}d t \right )}{2} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (t \right )=\frac {\mathit {Heaviside}\left (t \right )}{3}+\frac {{\mathrm e}^{-3 t} \mathit {Heaviside}\left (t \right )}{6}-\frac {\mathit {Heaviside}\left (t -1\right )}{3}-\frac {\mathit {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\mathit {Heaviside}\left (t -2\right )}{3}+\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\mathit {Heaviside}\left (t -3\right )}{3}-\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}+\frac {{\mathrm e}^{-t +1} \mathit {Heaviside}\left (t -1\right )}{2}-\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\mathit {Heaviside}\left (t \right ) {\mathrm e}^{-t}}{2} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{-t}+\frac {\mathit {Heaviside}\left (t \right )}{3}+\frac {{\mathrm e}^{-3 t} \mathit {Heaviside}\left (t \right )}{6}-\frac {\mathit {Heaviside}\left (t -1\right )}{3}-\frac {\mathit {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\mathit {Heaviside}\left (t -2\right )}{3}+\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\mathit {Heaviside}\left (t -3\right )}{3}-\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}+\frac {{\mathrm e}^{-t +1} \mathit {Heaviside}\left (t -1\right )}{2}-\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\mathit {Heaviside}\left (t \right ) {\mathrm e}^{-t}}{2} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{-t}+\frac {\mathit {Heaviside}\left (t \right )}{3}+\frac {{\mathrm e}^{-3 t} \mathit {Heaviside}\left (t \right )}{6}-\frac {\mathit {Heaviside}\left (t -1\right )}{3}-\frac {\mathit {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\mathit {Heaviside}\left (t -2\right )}{3}+\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\mathit {Heaviside}\left (t -3\right )}{3}-\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}+\frac {{\mathrm e}^{-t +1} \mathit {Heaviside}\left (t -1\right )}{2}-\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\mathit {Heaviside}\left (t \right ) {\mathrm e}^{-t}}{2} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=-\frac {2}{3} \\ {} & {} & -\frac {2}{3}=c_{1} +c_{2} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=-3 c_{1} {\mathrm e}^{-3 t}-c_{2} {\mathrm e}^{-t}+\frac {\mathit {Dirac}\left (t \right )}{3}-\frac {{\mathrm e}^{-3 t} \mathit {Heaviside}\left (t \right )}{2}+\frac {{\mathrm e}^{-3 t} \mathit {Dirac}\left (t \right )}{6}-\frac {\mathit {Dirac}\left (t -1\right )}{3}-\frac {\mathit {Dirac}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\mathit {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{2}+\frac {\mathit {Dirac}\left (t -2\right )}{3}+\frac {\mathit {Dirac}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{2}-\frac {\mathit {Dirac}\left (t -3\right )}{3}-\frac {\mathit {Dirac}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{2}+\frac {\mathit {Dirac}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}-\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}-\frac {{\mathrm e}^{-t +1} \mathit {Heaviside}\left (t -1\right )}{2}+\frac {{\mathrm e}^{-t +1} \mathit {Dirac}\left (t -1\right )}{2}-\frac {\mathit {Dirac}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}+\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\mathit {Dirac}\left (t \right ) {\mathrm e}^{-t}}{2}+\frac {\mathit {Heaviside}\left (t \right ) {\mathrm e}^{-t}}{2} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{t \hiderel {=}0\right \}}}}=1 \\ {} & {} & 1=-3 c_{1} -c_{2} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =-\frac {1}{6}, c_{2} =-\frac {1}{2}\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {\mathit {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {{\mathrm e}^{-t +1} \mathit {Heaviside}\left (t -1\right )}{2}-\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}+\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}-\frac {\mathit {Heaviside}\left (t -3\right )}{3}+\frac {\mathit {Heaviside}\left (t -2\right )}{3}-\frac {\mathit {Heaviside}\left (t -1\right )}{3}+\frac {\left (\mathit {Heaviside}\left (t \right )-1\right ) {\mathrm e}^{-3 t}}{6}+\frac {\left (-3 \mathit {Heaviside}\left (t \right )-3\right ) {\mathrm e}^{-t}}{6}+\frac {\mathit {Heaviside}\left (t \right )}{3} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {\mathit {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {{\mathrm e}^{-t +1} \mathit {Heaviside}\left (t -1\right )}{2}-\frac {\mathit {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}+\frac {\mathit {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}-\frac {\mathit {Heaviside}\left (t -3\right )}{3}+\frac {\mathit {Heaviside}\left (t -2\right )}{3}-\frac {\mathit {Heaviside}\left (t -1\right )}{3}+\frac {\left (\mathit {Heaviside}\left (t \right )-1\right ) {\mathrm e}^{-3 t}}{6}+\frac {\left (-3 \mathit {Heaviside}\left (t \right )-3\right ) {\mathrm e}^{-t}}{6}+\frac {\mathit {Heaviside}\left (t \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`

\[ y \left (t \right ) = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -3\right )}{3}-{\mathrm e}^{-t}-\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-3 t +9}}{6}+\frac {\operatorname {Heaviside}\left (t -3\right ) {\mathrm e}^{-t +3}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{6-3 t}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{2-t}}{2}+\frac {\operatorname {Heaviside}\left (t -2\right )}{3}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{6}+\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-t +1}}{2}-\frac {\operatorname {Heaviside}\left (t -1\right )}{3} \]

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{3}-e^{-t} & 0\leq t\leq 1 \\ -\frac {1}{6} e^{-3 t} \left (1+3 e^{2 t}\right ) & t<0 \\ \frac {1}{6} e^{-3 t} \left (-e^3-6 e^{2 t}+3 e^{2 t+1}\right ) & 1

4.24 problem Problem 3(j)

4.24.1 Existence and uniqueness analysis

4.24.2 Maple step by step solution