Forming differential equation in economic growth

Question

I have the following equations:

$ X = {\sqrt K} {\sqrt L} $

$ K' = 0.4X $

$ L = e^{0.04t} $

And my task is to form a differential equation and separate it.

I have come up with this formula: $ K' = 0.4e^{0.2t} * {\sqrt K} $

But this formula is presented in the solutions: $ K' = 0.4e^{0.02t} * {\sqrt K} $

Can somebody please explain how can $ {\sqrt{ e^{0.04t}}} $ become $ e^{0.02t} $ and not $ e^{0.2t} $ because $ {\sqrt{ 0.04}} $ is $ 0.2 $

Answer 1

Your actual question has nothing to do with ODEs. Note that $(a^b)^c=a^{bc}$ so

$$\sqrt{e^{0.04 t}}=(e^{4 t/100})^{1/2}=e^{2 t/100}=e^{0.02 t}$$

in agreement with the solutions.