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Zurtex

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My Calculus lecturer is pretty bad, so I just wanted to check over some stuff. First of all what use are Laplace transformations? Secondly he applied a laplace transformation to:

[tex]\frac{d^2x}{dt^2} - 5\frac{dx}{dt} + 6x = 3H(t) - 3H(t - 6)[/tex]

Where H(t) is the heavy side step function. However I really don't understand what went on there, could some one please explain how you at least start this?

Finally he defined the dirac delta function as being, 0 everywhere, except at a and that:

[tex]\int_{-\infty}^{\infty} \delta (x) dx= 1[/tex]

Does that make any sense at all?

[tex]\frac{d^2x}{dt^2} - 5\frac{dx}{dt} + 6x = 3H(t) - 3H(t - 6)[/tex]

Where H(t) is the heavy side step function. However I really don't understand what went on there, could some one please explain how you at least start this?

Finally he defined the dirac delta function as being, 0 everywhere, except at a and that:

[tex]\int_{-\infty}^{\infty} \delta (x) dx= 1[/tex]

Does that make any sense at all?

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