66.19.2 problem 28

Internal problem ID [16247]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 6. Laplace transform. Section 6.3 page 600
Problem number : 28
Date solved : Thursday, October 02, 2025 at 10:44:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y&={\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: 0.094 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)-4*y(t) = exp(2*t); 
ic:=[y(0) = 1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {13 \,{\mathrm e}^{-2 t}}{16}+\frac {{\mathrm e}^{2 t} \left (3+4 t \right )}{16} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 27

ode=D[y[t],{t,2}]-4*y[t]==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 \frac {1}{16} e^{-2 t} \left (e^{4 t} (4 t+3)+13\right ) \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*y(t) - 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)
 

\[ y{\left (t \right )} = \left (\frac {t}{4} + \frac {3}{16}\right ) e^{2 t} + \frac {13 e^{- 2 t}}{16} \]