76.13.37 problem 37

Internal problem ID [17548]
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 : 37
Date solved : Monday, March 31, 2025 at 04:17:07 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=2\\ y^{\prime }\left (-1\right )&=1 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 0; 
ic:=y(-1) = 2, D(y)(-1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (5 x +7\right ) {\mathrm e}^{-2-2 x} \]

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

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

\[ y(x)\to e^{-2 (x+1)} (5 x+7) \]

✓ Sympy. Time used: 0.158 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(-1): 2, Subs(Derivative(y(x), x), x, -1): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {5 x}{e^{2}} + \frac {7}{e^{2}}\right ) e^{- 2 x} \]