40.6.9 problem 9

Internal problem ID [8641]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.2, page 216
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:40:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+3 y&=6 t -8 \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: 0.104 (sec). Leaf size: 16

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+3*y(t) = 6*t-8; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 t -{\mathrm e}^{3 t}+{\mathrm e}^{t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 19

ode=D[y[t],{t,2}]-4*D[y[t],t]+3*y[t]==6*t-8; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 t+e^t-e^{3 t} \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*t + 3*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 8,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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