70.13.38 problem 38

Internal problem ID [18907]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 38
Date solved : Thursday, October 02, 2025 at 03:32:43 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+6 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: 0.115 (sec). Leaf size: 35

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+3*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\left (3+\sqrt {6}\right ) x} \left (\left (2+\sqrt {6}\right ) {\mathrm e}^{2 x \sqrt {6}}-\sqrt {6}+2\right )}{4} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 48

ode=D[y[x],{x,2}]+6*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]
 

\begin{align*} y(x)&\to \frac {1}{4} e^{-\left (\left (3+\sqrt {6}\right ) x\right )} \left (\left (2+\sqrt {6}\right ) e^{2 \sqrt {6} x}+2-\sqrt {6}\right ) \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*y(x) + 6*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)
 

\[ y{\left (x \right )} = \left (\frac {1}{2} + \frac {\sqrt {6}}{4}\right ) e^{x \left (-3 + \sqrt {6}\right )} + \left (\frac {1}{2} - \frac {\sqrt {6}}{4}\right ) e^{- x \left (\sqrt {6} + 3\right )} \]