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?

Question

sillybilly123 · Answer

$ 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:

$y = 9e^{ - 5t}$

celticcat · Answer

THanks