14.21.15 problem Problem 15

Internal problem ID [3942]
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 15
Date solved : Tuesday, September 30, 2025 at 06:59:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y&=12 \,{\mathrm e}^{2 t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.120 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)-y(t) = 12*exp(2*t); 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -5 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-t}+4 \,{\mathrm e}^{2 t} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 25

ode=D[y[t],{t,2}]-y[t]==12*Exp[2*t]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 e^{-t}-5 e^t+4 e^{2 t} \end{align*}

✓ Sympy. Time used: 0.051 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - 12*exp(2*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 4 e^{2 t} - 5 e^{t} + 2 e^{- t} \]