Ordinary Differential Equations

Theorem 3.1.4 shows that the Laplace transform of a sum is a sum of Laplace transforms. One might wonder if the Laplace transform of a product is the product of Laplace transforms. Unfortunately, that simply isn’t true (see if you can figure out why)! However, there is a special type of function "multiplication", called a convolution, whose Laplace transform is a product of Laplace transforms.

Definition 3.5.1. Convolution.

Let \(f\) and \(g\) be piecewise continuous on \(t\geq 0\text{.}\) The convolution of \(f\) and \(g\text{,}\) written \(f*g\) is defined by

\begin{equation*} (f*g)(t)=\displaystyle\int_0^t f(\tau)g(t-\tau)\mathrm{d}\tau. \end{equation*}

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Example 3.5.2.

If \(f(t)=t^2\) and \(g(t)=t\text{,}\) then compute \((f*g)(t)\text{.}\)

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Theorem 3.5.3. Properties of convolution.

Let \(f\text{,}\) \(g\text{,}\) and \(h\) be piecewise continuous on \(t \geq 0\text{.}\) Then,

1. \(f*g=g*f\text{,}\)
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2. \(f*(g+h)=(f*g)+(f*h)\text{,}\)
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3. \((f*g)*h=f*(g*h)\text{,}\) and
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4. \(f*0=0\text{.}\)
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Proof.

later!

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Theorem 3.5.4. Convolution theorem.

Let \(f\text{,}\) \(g\text{,}\) and \(h\) be piecewise continuous on \(t\geq 0\) and of exponential order. Then,

\begin{equation*} \mathcal{L}\{f*g\}(s)=\mathcal{L}\{f\}(s) \mathcal{L}\{g\}(s) \equiv F(s)G(s). \end{equation*}

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Example 3.5.5.

Take the Laplace transform of...

1. \(\displaystyle f(t)=t*\cos(t)\)
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2. \(\displaystyle g(t)=\displaystyle\int_0^t e^{\tau}\sin(t-\tau)\mathrm{d}\tau\)
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3. \(\displaystyle h(t)=e^{-t}*t^2e^t\)
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4. \(\displaystyle i(t)=\displaystyle\int_0^t \cos\left(3(t-\tau)\right)\mathrm{d}\tau\)
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Definition 3.5.6. Duhamel’s principle.

Consider the spring-mass equation \(ay''+by+cy=g(t), y(0)=y'(0)=0\text{.}\) Recall the transfer function

\begin{equation*} H(s)=\dfrac{Y(s)}{G(s)}=\dfrac{1}{as^2+bs+c}. \end{equation*}

Its corresponding inverse \(h(t)=\mathcal{L}^{-1}\{H\}(t)\) is called the weight of the system. Then the convolution \(\displaystyle\int_0^t h(\tau)g(t-\tau)\mathrm{d}\tau\) is called the Duhamel’s principle of the system.

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Example 3.5.7.

Let \(y''+6y'+10y=g(t)\text{,}\) where \(g\) is arbitrary but piecewise continuous on \(t \geq 0\text{.}\) Express \(y\) using Duhamel’s principle.

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Definition 3.5.8. Integro-differential equations.

A first-order integro-differential equation takes the form

\begin{equation*} y'(t)+\displaystyle\int_0^t y(\tau)g(t-\tau)\mathrm{d}\tau=h(t), \end{equation*}

where \(g\) and \(h\) are known functions.

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Integro-differential equations are useful when hte position/response variable is written in terms of itself. This is partiuclarly true for LRC circuits.

Example 3.5.9.

Recall that for a single-loop or series circuit, Kirchoff’s second law states that the sum of the voltage drops across an indcutor, resistor, and capacitor is equal to the impressed voltage \(E(t)\text{.}\) Then the current \(i(t)\) is governed by \(Li'(t)+Ri(t)+\dfrac{1}{C} \displaystyle\int_0^t i(\tau)\mathrm{d}\tau=E(t)\text{.}\)

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Example 3.5.10.

Solve the integro-differential equation

\begin{equation*} y'(t)+\displaystyle\int_0^t e^{-2\tau}y(t-\tau)\mathrm{d}\tau=1, y(0)=1. \end{equation*}

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Example 3.5.11.

Find the current \(i(t)\) in a single-loop LRC circuit where \(L=1 \text{h}\text{,}\) \(R=4 \Omega\text{,}\) \(C=0.25 f\text{,}\) \(i(0)=0\text{,}\) and the impressed voltage is \(E(t)=e^{-2t}\text{.}\)

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