76.13.31 problem 31

Internal problem ID [17542]
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 : 31
Date solved : Monday, March 31, 2025 at 04:16:55 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 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: 0.074 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+5*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 18

ode=D[y[x],{x,2}]+4*D[y[x],x]+5*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-2 x} (2 \sin (x)+\cos (x)) \]

✓ Sympy. Time used: 0.172 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) + 4*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)
 

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