2.9.11 problem 22

Internal problem ID [855]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.3, second order linear equations. Page 323
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 04:16:12 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 y^{\prime \prime }+6 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.095 (sec). Leaf size: 31

ode:=9*diff(diff(y(x),x),x)+6*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(0) = 3, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{3}} \left (5 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{3}\right )+3 \cos \left (\frac {\sqrt {3}\, x}{3}\right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 39

ode=9*D[y[x],{x,2}]+6*D[y[x],x]+4*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x/3} \left (5 \sqrt {3} \sin \left (\frac {x}{\sqrt {3}}\right )+3 \cos \left (\frac {x}{\sqrt {3}}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 6*Derivative(y(x), x) + 9*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (5 \sqrt {3} \sin {\left (\frac {\sqrt {3} x}{3} \right )} + 3 \cos {\left (\frac {\sqrt {3} x}{3} \right )}\right ) e^{- \frac {x}{3}} \]