76.13.41 problem 41

Internal problem ID [17552]
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 : 41
Date solved : Monday, March 31, 2025 at 04:17:13 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }-9 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+8*diff(y(x),x)-9*y(x) = 0; 
ic:=y(1) = 1, D(y)(1) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{9-9 x}}{10}+\frac {9 \,{\mathrm e}^{-1+x}}{10} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 26

ode=D[y[x],{x,2}]+8*D[y[x],x]-9*y[x]==0; 
ic={y[1]==1,Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{10} e^{9-9 x}+\frac {9 e^{x-1}}{10} \]

✓ Sympy. Time used: 0.170 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-9*y(x) + 8*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {9 e^{x}}{10 e} + \frac {e^{9} e^{- 9 x}}{10} \]