Question

I don't understand this problem:
Suppose y(t) = 9e - 5t is a solution of the initial value problem y' + ky = 0 what is the value of k?

Answer 1

\( y' + ky = 0\) is separable. As follows:

\( y' = - ky \)

\(\dfrac{ y'}{y} = - k \)

So integrate:

\(\int \dfrac{ dy}{y} = - k \int dx\)

\(\implies \ln y = - k x + C\)

Take logs both sides:

\(y = e^{- k x + C} = e^{- k x } e^C = A e^{- kx}\)

We've just played with the integration constant.


Then pattern match back to your solution:

\(y = 9e^{ - 5t}\)

Answer 2

THanks