(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve using the Laplace Transforms (can not use partial fractions)

f '(t) + [tex]\int2f(u) du[/tex] = 2 + 3f(t)

2. Relevant equations

Using Laplace

f '(t) gets replaced with sF(s) -f(0)

[tex]\int2f(u) du[/tex] gets replaced with [tex]\frac{2F(s)}{s}[/tex]

Please correct me if I'm wrong on the replacements here.

3. The attempt at a solution

After using Laplace on both sides I get

[tex]sF(s)-f(0)+\frac{2F(s)}{s} = \frac{2}{s} + 3F(s)[\tex]

[tex]sF(s)-3F(s)+\frac{2F(s)}{s} = \frac{2}{s} + f(0)[\tex]

[tex]F(S)(s-3+\frac{2}{s}) = \frac{2}{s} + f(0)[\tex]

Divide through and manipulate a little to get:

[tex]F(S) = \frac{2}{(s-2)(s-1)} + f(0)\frac{s}{(s-2)(s-1)}[\tex]

OK, here is where I get stuck. The first half I can figure out, it's the s/((s-2)(s-1)) that I can't figure out. I did find a transform in the Laplace tables in the back of the book but this particular transform was not on the list of approved transforms we could use freely (without proving).

So, I've either screwed up in my math here somewhere's or I have to prove the Inverse Laplace Transform of s/((s-2)(s-1)). Now someone mentioned using the L'Hopitals rule on it but I don't see how.

I'm completely at a loss. Please help.

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# Homework Help: Laplace Transform, what am I doing wrong?

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