33.5.11 problem Problem 24.33

Internal problem ID [7840]
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 : Tuesday, September 30, 2025 at 05:06:11 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*}

✓ Maple. Time used: 0.197 (sec). Leaf size: 45

ode:=diff(diff(y(x),x),x)+5*diff(y(x),x)-3*y(x) = Heaviside(x-4); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

\[ y = \frac {\operatorname {Heaviside}\left (x -4\right ) \left (-37+{\mathrm e}^{10-\frac {5 x}{2}} \left (5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}+37 \cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right )\right )\right )}{111} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 70

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

\begin{align*} 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} \end{align*}

✓ Sympy. Time used: 0.785 (sec). Leaf size: 136

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*y(x) - Heaviside(x - 4) + 5*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {2 \sqrt {37} e^{\frac {x \left (-5 + \sqrt {37}\right )}{2}} \theta \left (x - 4\right )}{37 \left (-5 + \sqrt {37}\right ) e^{-10 + 2 \sqrt {37}}} + \frac {2 \sqrt {37} e^{- \frac {\sqrt {37} x}{2} - \frac {5 x}{2} + 10 + 2 \sqrt {37}} \theta \left (x - 4\right )}{37 \left (5 + \sqrt {37}\right )} - \frac {2 \sqrt {37} \theta \left (x - 4\right )}{37 \left (-5 + \sqrt {37}\right )} - \frac {2 \sqrt {37} \theta \left (x - 4\right )}{37 \left (5 + \sqrt {37}\right )} \]