52.6.5 problem 25

Internal problem ID [8340]
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 : 25
Date solved : Wednesday, March 05, 2025 at 05:35:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=t \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: 0.556 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-6*diff(y(t),t)+9*y(t) = t; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 25

ode=D[y[t],{t,2}]-6*D[y[t],t]+9*y[t]==t; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{27} \left (3 t+e^{3 t} (30 t-2)+2\right ) \]

✓ Sympy. Time used: 0.224 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 9*y(t) - 6*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)
 

\[ y{\left (t \right )} = \frac {t}{9} + \left (\frac {10 t}{9} - \frac {2}{27}\right ) e^{3 t} + \frac {2}{27} \]