5.11 problem Problem 24.33

Internal problem ID [5210]
Internal file name [OUTPUT/4703_Sunday_June_05_2022_03_03_40_PM_78854413/index.tex]

Book: Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section: Chapter 24. Solutions of linear DE by Laplace transforms. Supplementary Problems. page 248
Problem number: Problem 24.33.
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]]

\[ \boxed {y^{\prime \prime }+5 y^{\prime }-3 y=\operatorname {Heaviside}\left (x -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}

5.11.1 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) &=5\\ q(x) &=-3\\ F &=\operatorname {Heaviside}\left (x -4\right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }+5 y^{\prime }-3 y = \operatorname {Heaviside}\left (x -4\right ) \end {align*}

The domain of \(p(x)=5\) 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*} s^{2} Y \left (s \right )-y^{\prime }\left (0\right )-s y \left (0\right )+5 s Y \left (s \right )-5 y \left (0\right )-3 Y \left (s \right ) = \frac {{\mathrm e}^{-4 s}}{s}\tag {1} \end {align*}

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

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

Solving the above equation for \(Y(s)\) results in \begin {align*} Y(s) = \frac {{\mathrm e}^{-4 s}}{s \left (s^{2}+5 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 {{\mathrm e}^{-4 s}}{s \left (s^{2}+5 s -3\right )}\right )\\ &= \frac {\operatorname {Heaviside}\left (x -4\right ) \left (-37+{\mathrm e}^{10-\frac {5 x}{2}} \left (5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}+37 \cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right )\right )\right )}{111} \end {align*}

Converting the above solution to piecewise it becomes \[ y = \left \{\begin {array}{cc} 0 & x <4 \\ -\frac {1}{3}+\frac {{\mathrm e}^{10-\frac {5 x}{2}} \left (5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}+37 \cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right )\right )}{111} & 4\le x \end {array}\right . \] Simplifying the solution gives \[ y = \left \{\begin {array}{cc} 0 & x <4 \\ \frac {5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}\, {\mathrm e}^{10-\frac {5 x}{2}}}{111}+\frac {\cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) {\mathrm e}^{10-\frac {5 x}{2}}}{3}-\frac {1}{3} & 4\le x \end {array}\right . \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \left \{\begin {array}{cc} 0 & x <4 \\ \frac {5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}\, {\mathrm e}^{10-\frac {5 x}{2}}}{111}+\frac {\cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) {\mathrm e}^{10-\frac {5 x}{2}}}{3}-\frac {1}{3} & 4\le x \end {array}\right . \\ \end{align*}

Verification of solutions

\[ y = \left \{\begin {array}{cc} 0 & x <4 \\ \frac {5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}\, {\mathrm e}^{10-\frac {5 x}{2}}}{111}+\frac {\cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) {\mathrm e}^{10-\frac {5 x}{2}}}{3}-\frac {1}{3} & 4\le x \end {array}\right . \] Verified OK.

5.11.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime \prime }+5 y^{\prime }-3 y=\mathit {Heaviside}\left (x -4\right ), y \left (0\right )=0, y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0\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}+5 r -3=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {\left (-5\right )\pm \left (\sqrt {37}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}, -\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) \\ \bullet & {} & \textrm {1st solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (x \right )={\mathrm e}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x} \\ \bullet & {} & \textrm {2nd solution of the homogeneous ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (x \right )={\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) 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}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x}+c_{2} {\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) 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 )=\mathit {Heaviside}\left (x -4\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}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x} & {\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) x} \\ \left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x} & \left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) x} \end {array}\right ] \\ {} & \circ & \textrm {Compute Wronskian}\hspace {3pt} \\ {} & {} & W \left (y_{1}\left (x \right ), y_{2}\left (x \right )\right )=\sqrt {37}\, {\mathrm e}^{-5 x} \\ {} & \circ & \textrm {Substitute functions into equation for}\hspace {3pt} y_{p}\left (x \right ) \\ {} & {} & y_{p}\left (x \right )=-\frac {\sqrt {37}\, \left ({\mathrm e}^{-\frac {\left (5+\sqrt {37}\right ) x}{2}} \left (\int {\mathrm e}^{\frac {\left (5+\sqrt {37}\right ) x}{2}} \mathit {Heaviside}\left (x -4\right )d x \right )-{\mathrm e}^{\frac {\left (-5+\sqrt {37}\right ) x}{2}} \left (\int {\mathrm e}^{-\frac {\left (-5+\sqrt {37}\right ) x}{2}} \mathit {Heaviside}\left (x -4\right )d x \right )\right )}{37} \\ {} & \circ & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & y_{p}\left (x \right )=-\frac {\mathit {Heaviside}\left (x -4\right ) \left (5 \sqrt {37}\, {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}+74\right )}{222} \\ \bullet & {} & \textrm {Substitute particular solution into general solution to ODE}\hspace {3pt} \\ {} & {} & y=c_{1} {\mathrm e}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x}+c_{2} {\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) x}-\frac {\mathit {Heaviside}\left (x -4\right ) \left (5 \sqrt {37}\, {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}+74\right )}{222} \\ \square & {} & \textrm {Check validity of solution}\hspace {3pt} y=c_{1} {\mathrm e}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x}+c_{2} {\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) x}-\frac {\mathit {Heaviside}\left (x -4\right ) \left (5 \sqrt {37}\, {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}+74\right )}{222} \\ {} & \circ & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=0 \\ {} & {} & 0=c_{1} +c_{2} \\ {} & \circ & \textrm {Compute derivative of the solution}\hspace {3pt} \\ {} & {} & y^{\prime }=c_{1} \left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{\left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) x}+c_{2} \left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{\left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) x}-\frac {\mathit {Dirac}\left (x -4\right ) \left (5 \sqrt {37}\, {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}+74\right )}{222}-\frac {\mathit {Heaviside}\left (x -4\right ) \left (5 \sqrt {37}\, \left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, \left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 \left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 \left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}\right )}{222} \\ {} & \circ & \textrm {Use the initial condition}\hspace {3pt} y^{\prime }{\raise{-0.36em}{\Big |}}{\mstack {}{_{\left \{x \hiderel {=}0\right \}}}}=0 \\ {} & {} & 0=c_{1} \left (-\frac {5}{2}-\frac {\sqrt {37}}{2}\right )+c_{2} \left (-\frac {5}{2}+\frac {\sqrt {37}}{2}\right ) \\ {} & \circ & \textrm {Solve for}\hspace {3pt} c_{1} \hspace {3pt}\textrm {and}\hspace {3pt} c_{2} \\ {} & {} & \left \{c_{1} =0, c_{2} =0\right \} \\ {} & \circ & \textrm {Substitute constant values into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {\mathit {Heaviside}\left (x -4\right ) \left (5 \sqrt {37}\, {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}+74\right )}{222} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {\mathit {Heaviside}\left (x -4\right ) \left (5 \sqrt {37}\, {\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-5 \sqrt {37}\, {\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{-\frac {\left (x -4\right ) \left (5+\sqrt {37}\right )}{2}}-37 \,{\mathrm e}^{\frac {\left (x -4\right ) \left (-5+\sqrt {37}\right )}{2}}+74\right )}{222} \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: 0.531 (sec). Leaf size: 45

dsolve([diff(y(x),x$2)+5*diff(y(x),x)-3*y(x)=Heaviside(x-4),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\operatorname {Heaviside}\left (x -4\right ) \left (-1+\frac {5 \sqrt {37}\, \sinh \left (\frac {\left (x -4\right ) \sqrt {37}}{2}\right ) {\mathrm e}^{-\frac {5 x}{2}+10}}{37}+\cosh \left (\frac {\left (x -4\right ) \sqrt {37}}{2}\right ) {\mathrm e}^{-\frac {5 x}{2}+10}\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 70

DSolve[{y''[x]+5*y'[x]-3*y[x]==UnitStep[x-4],{y[0]==0,y'[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{222} \left (-74+\left (37+5 \sqrt {37}\right ) e^{\frac {1}{2} \left (-5+\sqrt {37}\right ) (x-4)}+\left (37-5 \sqrt {37}\right ) e^{-\frac {1}{2} \left (5+\sqrt {37}\right ) (x-4)}\right ) & x>4 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]