10.7.13 problem 13

Internal problem ID [1261]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.1 Homogeneous Equations with Constant Coefficients, page 144
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 12:26:15 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+3 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: 0.039 (sec). Leaf size: 41

ode:=diff(diff(y(x),x),x)+5*diff(y(x),x)+3*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (13+5 \sqrt {13}\right ) {\mathrm e}^{\frac {\left (-5+\sqrt {13}\right ) x}{2}}}{26}+\frac {\left (13-5 \sqrt {13}\right ) {\mathrm e}^{-\frac {\left (5+\sqrt {13}\right ) x}{2}}}{26} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 51

ode=D[y[x],{x,2}]+5*D[y[x],x]+3*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{26} e^{-\frac {1}{2} \left (5+\sqrt {13}\right ) x} \left (\left (13+5 \sqrt {13}\right ) e^{\sqrt {13} x}+13-5 \sqrt {13}\right ) \]

✓ Sympy. Time used: 0.251 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*y(x) + 5*Derivative(y(x), x) + 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)
 

\[ y{\left (x \right )} = \left (\frac {1}{2} + \frac {5 \sqrt {13}}{26}\right ) e^{\frac {x \left (-5 + \sqrt {13}\right )}{2}} + \left (\frac {1}{2} - \frac {5 \sqrt {13}}{26}\right ) e^{- \frac {x \left (\sqrt {13} + 5\right )}{2}} \]