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

`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)`.

Answer 2

m = -6

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