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# Laplace transform of squared Function and a multiple of several functions?

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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

Oct 4, 2017
edited by Quazars  Oct 4, 2017

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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.

Oct 4, 2017
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Ah I see, competely went over my head lol. Still I'm wondering if it is possible solving with Laplace Transforms that's all. How does it apply to squared functions! Thanks

Quazars  Oct 4, 2017
edited by Quazars  Oct 4, 2017