I stumbled upon the following differential equation:

\(y'=k_1(k_2-y)y\)

I've never encountered a squared function before so I'm kind of stumped on how to proceed with this. I tried putting the following into Wolfram Alpha:

\(\mathscr{L} \{(f(t))^2 \}\)

Which gave me the following error:

(no result found in terms of standard mathematical functions)

So I could use some help with that as Googling didn't give me what I was looking for. On the topic I was also curious about rules for laplace transform of two functions multiplied, I know that the laplace transform is a linear operator but maybe if there's some sort of general rule that applies here:

\(\mathscr{L} \{f(t) \cdot g(t) \}\)

I'm not expecting someone to spoonfeed me too much here but a pointer to some site or pdf/book would be nice. I'm not the most informed on Laplace Transforms though, so try to keep it somewhat simple :) Thanks

Quazars
Oct 4, 2017

#1**+2 **

Why Laplace Transforms ? Are you required to use this method for some reason ?

The natural method to use is a separation of variables.

\(\displaystyle \frac{dy}{dt}=k_{1}(k_{2}-y)y\), so, \(\displaystyle \frac{dy}{(k_{2}-y)y}=k_{1}dt\).

Split the lhs into partial fractions,

\(\displaystyle \frac{1}{k_{2}y}+\frac{1}{k_{2}(k_{2}-y)}\)

and integrate on both sides.

Tiggsy.

Guest Oct 4, 2017