2.7.12 problem 12

Internal problem ID [818]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.1, second order linear equations. Page 299
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 04:15:44 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+13*y(x) = 0; 
ic:=[y(0) = 2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \left (3 \sin \left (2 x \right )+2 \cos \left (2 x \right )\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 24

ode=D[y[x],{x,2}]+6*D[y[x],x]+13*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-3 x} (3 \sin (2 x)+2 \cos (2 x)) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (3 \sin {\left (2 x \right )} + 2 \cos {\left (2 x \right )}\right ) e^{- 3 x} \]