# Differential Equations Question

1. Jun 19, 2013

### Miike012

Can anyone tell me how the book arrived at the portion that I underlined in the paint document?

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2. Jun 20, 2013

### Office_Shredder

Staff Emeritus
It's using the fact that
$$\mathfrak{L}\left( \sum a_i \frac{d^k}{dx^k} y \right) = \sum a_i \mathfrak{L} \left( \frac{d^k}{dx^k} y \right)$$
And that if you take the Laplace operator of the kth derivative of y you get sk L(y) plus some values of y and its derivatives at 0 (more specifically the general differentiation property at http://en.wikipedia.org/wiki/Laplace_transform#Properties_and_theorems)

The ak coefficients that were next to the differential operators stick around and multiply the sk L(y) guys, meaning you get exactly q(s) out if you started with q(D), but the polynomial terms depend on the derivatives at zero so is hard to calculate what its relationship with q(s) is