10.8.19 problem 25

Internal problem ID [1291]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.3 Complex Roots of the Characteristic Equation , page 164
Problem number : 25
Date solved : Tuesday, March 04, 2025 at 12:28:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+6 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 32

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

\[ y = \frac {{\mathrm e}^{-x} \left (\sqrt {5}\, \left (\alpha +2\right ) \sin \left (\sqrt {5}\, x \right )+10 \cos \left (\sqrt {5}\, x \right )\right )}{5} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 42

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

\[ y(x)\to \frac {1}{5} e^{-x} \left (\sqrt {5} (\alpha +2) \sin \left (\sqrt {5} x\right )+10 \cos \left (\sqrt {5} x\right )\right ) \]

✓ Sympy. Time used: 0.171 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \left (\left (\frac {\sqrt {5} \alpha }{5} + \frac {2 \sqrt {5}}{5}\right ) \sin {\left (\sqrt {5} x \right )} + 2 \cos {\left (\sqrt {5} x \right )}\right ) e^{- x} \]