Question

Find all values of m so that the function y = emx is a solution of the given differential equation. (enter your answers as a comma-separated list.) y' + 6y = 0

Answer 1

Answer:

m = -6

Step-by-step explanation:

Suppose we have an differential equation in the following format:

ay' + by = 0

The differential equation has the following equivalent polynomial

ar + b = 0.

The root of the polynomial is a value of m so that the function y = e^(my) is a solution of the equations.

In this problem, we have that:

y' + 6y = 0

The polynomial is

r + 6 = 0

r = -6

So the value is m = -6

Answer 2

$y=e^{mx}\implies y'=me^{mx}$

$y'+6y=0\iff me^{mx}+6e^{mx}=(m+6)e^{mx}=0\implies m=-6$

So one solution to the ODE is $y=e^{-6x}$.