52.6.18 problem 68

Internal problem ID [8353]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 68
Date solved : Wednesday, March 05, 2025 at 05:35:41 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&=\operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.694 (sec). Leaf size: 45

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)+6*y(t) = Heaviside(t-1); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 60

ode=D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==UnitStep[t-1]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.718 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(6*y(t) - Heaviside(t - 1) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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