67.3.2 problem Problem 3

Internal problem ID [13949]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 3
Date solved : Monday, March 31, 2025 at 08:20:05 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 y^{\prime \prime }-4 y^{\prime }+5 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.109 (sec). Leaf size: 15

ode:=4*diff(diff(y(t),t),t)-4*diff(y(t),t)+5*y(t) = 0; 
ic:=y(0) = 2, D(y)(0) = 3; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 19

ode=4*D[y[t],{t,2}]-4*D[y[t],t]+5*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to 2 e^{t/2} (\sin (t)+\cos (t)) \]

✓ Sympy. Time used: 0.211 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

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