46.6.8 problem 28

Internal problem ID [9629]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 7 THE LAPLACE TRANSFORM. 7.3.1 TRANSLATION ON THE s-AXIS. Page 297
Problem number : 28
Date solved : Tuesday, September 30, 2025 at 06:21:37 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime }+20 y^{\prime }+51 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: 0.122 (sec). Leaf size: 30

ode:=2*diff(diff(y(t),t),t)+20*diff(y(t),t)+51*y(t) = 0; 
ic:=[y(0) = 2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \,{\mathrm e}^{-5 t} \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+5 \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 36

ode=2*D[y[t],{t,2}]+20*D[y[t],t]+51*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 2 e^{-5 t} \left (5 \sqrt {2} \sin \left (\frac {t}{\sqrt {2}}\right )+\cos \left (\frac {t}{\sqrt {2}}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(51*y(t) + 20*Derivative(y(t), t) + 2*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)
 

\[ y{\left (t \right )} = \left (10 \sqrt {2} \sin {\left (\frac {\sqrt {2} t}{2} \right )} + 2 \cos {\left (\frac {\sqrt {2} t}{2} \right )}\right ) e^{- 5 t} \]