14.8.4 problem 4

Internal problem ID [2559]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.1 Linear equations with constant coefficients (complex roots). Excercises page 144
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:10:10 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

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

\[ y = {\mathrm e}^{\frac {t}{8}} \left (c_1 \sin \left (\frac {\sqrt {15}\, t}{8}\right )+c_2 \cos \left (\frac {\sqrt {15}\, t}{8}\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 42

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

\[ y(t)\to e^{t/8} \left (c_2 \cos \left (\frac {\sqrt {15} t}{8}\right )+c_1 \sin \left (\frac {\sqrt {15} t}{8}\right )\right ) \]

✓ Sympy. Time used: 0.171 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {15} t}{8} \right )} + C_{2} \cos {\left (\frac {\sqrt {15} t}{8} \right )}\right ) e^{\frac {t}{8}} \]