84.38.13 problem 26.25

Internal problem ID [22367]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Supplementary problems
Problem number : 26.25
Date solved : Thursday, October 02, 2025 at 08:37:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }-3 y&=\operatorname {Heaviside}\left (-4+t \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: 0.056 (sec). Leaf size: 45

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)-3*y(t) = Heaviside(t-4); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 70

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

\begin{align*} y(t)&\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 ) (t-4)}+\left (37-5 \sqrt {37}\right ) e^{-\frac {1}{2} \left (5+\sqrt {37}\right ) (t-4)}\right ) & t>4 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.777 (sec). Leaf size: 136

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

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