is there a difference?

I have the equation:

$P(t)=\frac{K}{9e^{-rt}+1}$

this equation was derived from solving the logistic growth equation:

$\frac{dP}{dt}=rP\big(1-\frac{P}{K}\big)$

where,

r is the rate of growth = 1.13

K is the maximum population = 10,000

I've been asked to change the growth rate to:

$r = 0.83 + \frac{0.3}{1+0.3t}$

My question is,

Do I have to solve the entire logistic growth equation again, that is,

or can I just substituted r into the derived equation and rearrage, that is,

I wanted to get some clarification as I've tried doing both process with a simpler equation, however the answer still get quite complex and I'm not sure If I've done it correctly.

Thank you.