14.21.17 problem Problem 17

Internal problem ID [3944]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.4. page 689
Problem number : Problem 17
Date solved : Tuesday, September 30, 2025 at 06:59:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }-6 y&=12-6 \,{\mathrm e}^{t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.098 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-diff(y(t),t)-6*y(t) = 12-6*exp(t); 
ic:=[y(0) = 5, D(y)(0) = -3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {8 \,{\mathrm e}^{3 t}}{5}-2+{\mathrm e}^{t}+\frac {22 \,{\mathrm e}^{-2 t}}{5} \]

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 28

ode=D[y[t],{t,2}]-D[y[t],t]-6*y[t]==6*(2-Exp[t]); 
ic={y[0]==5,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {22 e^{-2 t}}{5}+e^t+\frac {8 e^{3 t}}{5}-2 \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*y(t) + 6*exp(t) - Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 12,0) 
ics = {y(0): 5, Subs(Derivative(y(t), t), t, 0): -3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {8 e^{3 t}}{5} + e^{t} - 2 + \frac {22 e^{- 2 t}}{5} \]