78.2.3 problem 1.c

Internal problem ID [20955]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 1.c
Date solved : Thursday, October 02, 2025 at 07:00:39 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 8 y^{\prime \prime }+4 y^{\prime }+y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.062 (sec). Leaf size: 18

ode:=8*diff(diff(y(x),x),x)+4*diff(y(x),x)+y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{4}} \left (\sin \left (\frac {x}{4}\right )+\cos \left (\frac {x}{4}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 26

ode=8*D[y[x],{x,2}]+4*D[y[x],x]+y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x/4} \left (\sin \left (\frac {x}{4}\right )+\cos \left (\frac {x}{4}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + 4*Derivative(y(x), x) + 8*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\sin {\left (\frac {x}{4} \right )} + \cos {\left (\frac {x}{4} \right )}\right ) e^{- \frac {x}{4}} \]