39.5.11 problem Problem 24.33

Internal problem ID [6553]
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
Date solved : Monday, January 27, 2025 at 02:12:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.232 (sec). Leaf size: 48

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)
 

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

✓ Solution by Mathematica

Time used: 0.056 (sec). Leaf size: 70

DSolve[{D[y[x],{x,2}]+5*D[y[x],x]-3*y[x]==UnitStep[x-4],{y[0]==0,Derivative[1][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

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