83.21.2 problem 2

Internal problem ID [22062]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination VIII. page 256
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:23:18 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 20

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

\begin{align*} y(t)&\to e^{-t} (\sin (2 t)+\cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {3 \sqrt {19} \sin {\left (\frac {\sqrt {19} t}{2} \right )}}{19} + \cos {\left (\frac {\sqrt {19} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]