67.3.6 problem Problem 7

Internal problem ID [13953]
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 7
Date solved : Monday, March 31, 2025 at 08:20:10 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 y^{\prime \prime }-4 y^{\prime }+37 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: 0.164 (sec). Leaf size: 23

ode:=4*diff(diff(y(t),t),t)-4*diff(y(t),t)+37*y(t) = 0; 
ic:=y(0) = 2, D(y)(0) = -3; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 29

ode=4*D[y[t],{t,2}]-4*D[y[t],t]+37*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {2}{3} e^{t/2} (3 \cos (3 t)-2 \sin (3 t)) \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(37*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)
 

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