5.3.4 problem 11

Internal problem ID [1486]
Book : Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section : Chapter 6.2, The Laplace Transform. Solution of Initial Value Problems. page 255
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 04:34:33 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+4 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 27

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

\[ y = -\frac {2 \,{\mathrm e}^{t} \left (-3 \cos \left (\sqrt {3}\, t \right )+\sqrt {3}\, \sin \left (\sqrt {3}\, t \right )\right )}{3} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 37

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

\begin{align*} y(t)&\to -\frac {2}{3} e^t \left (\sqrt {3} \sin \left (\sqrt {3} t\right )-3 \cos \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 32

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

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