Ordinary Differential Equations

In previous sectrions, we have calculate the Laplace transform of derivatives. Now we calculate the derivative of the transform itself.

Theorem 3.6.1. Differentiation of Laplace transform.

Let \(f\) be piecewise continuous for \(t \geq 0\) and of exponential order. Then for \(n \in \{1,2,3,\ldots\}\text{,}\)

\begin{equation*} \mathcal{L}\{t^n f(t)\}(s) = (-1)^n F^{(n)}(s). \end{equation*}

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Proof.

later!

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Example 3.6.2.

Find the Laplace transform of...

1. \(\displaystyle f(t)=te^{3t}\)
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2. \(\displaystyle g(t)=t\cos(2t)\)
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Some inverse Laplace transforms are easier to calculate by taking the derivative of \(F\) first:

Example 3.6.3.

Find the inverse Laplace transform of...

1. \(\displaystyle F(s)=\arctan(s)\)
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2. \(\displaystyle G(s)=\ln\left(\dfrac{s+2}{s-2}\right)\)
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Example 3.6.4.

Use the Laplace transform to solve

\begin{equation*} y''+2ty'-4y=1, y(0)=0, y'(0)=0. \end{equation*}

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