Solving a Simple Laplace Transform Equation: Tips for Beginners

[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?

Homework Statement

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

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

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

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.

 

[Ray Vickson]

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?

Homework Statement

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

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

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

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.

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