7.8.3 problem 2

Internal problem ID [2378]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:35:05 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime }+3 y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

ode:=2*diff(diff(y(t),t),t)+3*diff(y(t),t)+4*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {3 t}{4}} \left (c_1 \sin \left (\frac {\sqrt {23}\, t}{4}\right )+c_2 \cos \left (\frac {\sqrt {23}\, t}{4}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 42

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

\begin{align*} y(t)&\to e^{-3 t/4} \left (c_2 \cos \left (\frac {\sqrt {23} t}{4}\right )+c_1 \sin \left (\frac {\sqrt {23} t}{4}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 32

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

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {23} t}{4} \right )} + C_{2} \cos {\left (\frac {\sqrt {23} t}{4} \right )}\right ) e^{- \frac {3 t}{4}} \]