40.6.5 problem 5

Internal problem ID [8637]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.2, page 216
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:40:05 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=12 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.086 (sec). Leaf size: 10

ode:=diff(diff(y(t),t),t)-1/4*y(t) = 0; 
ic:=[y(0) = 12, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 12 \cosh \left (\frac {t}{2}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 19

ode=D[y[t],{t,2}]-1/4*y[t]==0; 
ic={y[0]==12,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 6 e^{-t/2} \left (e^t+1\right ) \end{align*}

✓ Sympy. Time used: 0.048 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)/4 + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 12, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 6 e^{\frac {t}{2}} + 6 e^{- \frac {t}{2}} \]