61.3.15 problem Problem 16

Internal problem ID [15314]
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 16
Date solved : Thursday, October 02, 2025 at 10:11:33 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime }+3 y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 17

ode:=2*diff(diff(y(t),t),t)+3*diff(y(t),t)+y(t) = 0; 
ic:=[y(0) = 3, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -{\mathrm e}^{-t}+4 \,{\mathrm e}^{-\frac {t}{2}} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 22

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

\begin{align*} y(t)&\to e^{-t} \left (4 e^{t/2}-1\right ) \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + 3*Derivative(y(t), t) + 2*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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