14.21.12 problem Problem 12

Internal problem ID [3939]
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 12
Date solved : Tuesday, September 30, 2025 at 06:59:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=10 \,{\mathrm e}^{-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: 0.125 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-2*y(t) = 10*exp(-t); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+D[y[t],t]-2*y[t]==10*Exp[-t]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.105 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 10*exp(-t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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