Simple (?) Laplace Transform

I'm new to Laplace and having slight difficulty with what looks like an obvious equation. I can do basic first and second order equations. Why is it that the seemingly easy equations are always the ones to stump you?

1. The problem statement, all variables and given/known data

$\frac{dy}{dx}=-y$

$y=2, x=0, y_{0}=2$

2. Relevant equations

This equation can also be written $y'=-y$

A function table I have states that the transforms are:

$y'= s∠ {y}-y_{0}$

$y=\frac{y}{s}$ so $-y=\frac{-y}{s}$

3. The attempt at a solution

$s∠ {y}=\frac{-y}{s}+y_{0}$

$∠ {y}=\frac{-2}{s^{2}}+\frac{2}{s}$

A-ha!

$\frac{-2}{s{^2}}$ = $\frac{A}{s} + \frac{B}{s}$

So

$-2=As+Bs$

And I'm stumped. I can't eliminate A without eliminating B, and vice versa, so what is my next move? I don't see an error in my calculations, so I assume that I must be missing a key piece of knowledge regarding Laplace that will see me through. I think I remember reading that Laplace transforms are for non-homogeneous equations, although the terms 'homogeneous' and 'non-homogeneous' have always been greeted with *nomenclature overload - initialising file <happy_place>*.

Any help is much appreciated. I quite like these Laplace transforms, or did anyway.
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 Quote by AngusBurger I'm new to Laplace and having slight difficulty with what looks like an obvious equation. I can do basic first and second order equations. Why is it that the seemingly easy equations are always the ones to stump you? 1. The problem statement, all variables and given/known data $\frac{dy}{dx}=-y$ $y=2, x=0, y_{0}=2$ 2. Relevant equations This equation can also be written $y'=-y$ A function table I have states that the transforms are: $y'= s∠ {y}-y_{0}$ $y=\frac{y}{s}$ so $-y=\frac{-y}{s}$ 3. The attempt at a solution $s∠ {y}=\frac{-y}{s}+y_{0}$ $∠ {y}=\frac{-2}{s^{2}}+\frac{2}{s}$ A-ha! $\frac{-2}{s{^2}}$ = $\frac{A}{s} + \frac{B}{s}$ So $-2=As+Bs$ And I'm stumped. I can't eliminate A without eliminating B, and vice versa, so what is my next move? I don't see an error in my calculations, so I assume that I must be missing a key piece of knowledge regarding Laplace that will see me through. I think I remember reading that Laplace transforms are for non-homogeneous equations, although the terms 'homogeneous' and 'non-homogeneous' have always been greeted with *nomenclature overload - initialising file *. Any help is much appreciated. I quite like these Laplace transforms, or did anyway.
A lot of what you wrote is either incomprehensible or just plain wrong. For example, it is not true that $y=\frac{y}{s},$ and $s∠ {y}=\frac{-y}{s}+y_{0}$ is nonsense as well. If by the notation $s∠ {y}$ you mean $s Y(s),$ where Y(s) is the transform of y(t), then of course, the left-hand side is a function of s alone, while your right-hand-side has y in it! I suggest you start over again, and be more careful.

RGV

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