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Section4.4Exercises

1

Response to Exponential Input

Consider the following differential equation:

\begin{equation}\frac{dy}{dt} = ay + e^{st}\label{men-28}\tag{4.4.5}\end{equation}\begin{equation}y=y(0)\:at\:t=0\label{men-29}\tag{4.4.6}\end{equation}

This is the standard growth equation with an exponential forcing term \(e^{st}\).

We look for a particular solutions of the form:

\begin{equation*}y_p = Ye^{st}\end{equation*}

where \(Y\) is a constant.

Substituting we get:

\begin{equation*}Ys e^{st} = aY e^{st} + e^{st}\end{equation*}

Rearranging gives:

\begin{equation*}Y=\frac{1}{s-a}\end{equation*}

Full solution is the particular solution plus null (or homogeneous) solution:

\begin{equation*}y(t)=\frac{e^{st}}{s-a} + C e^{st}\end{equation*}

From the initial conditions we get:

\begin{equation*}y(0) = \frac{1}{s-a} + C\end{equation*}

Substituting in \(y(t)\) gives:

\begin{equation*}y(t)=\frac{e^{st}}{s-a} + \left[y(0) -\frac{1}{s-a}\right] e^{st}\end{equation*}

Rearranging we can see the effect of the forcing in addition to the homogeneous term:

\begin{equation*}y(t)= y(0) e^{st} + \frac{e^{st}-e^{at}}{s-a}\end{equation*}

In Strang's terminology:

\begin{equation}y(t) = y_p + y_n = y(0) e^{st} + \frac{e^{st}-e^{at}}{s-a}\label{men-30}\tag{4.4.7}\end{equation}

That is, the solution is a combination of the standard homogeneous term - the standard growth factor - plus an extra term coming from the forcing factor. Notice that this extra term has a singularity (division by zero) at \(s=a\). This is Resonance.

This singularity can be handled using L'Hopital's Rule4.0.3

Figure4.4.1Gilbert Strang In Action

The full solution, using L'Hopital, should match the final solution in Strang's Lecture.

\begin{equation}y(t)=y_{0} e^{a t} + t e^{s t}\label{men-31}\tag{4.4.8}\end{equation}

Now try and solve this using SageMath.

Hint Solution
2

Response to Oscillating Input

Consider the following differential equation:

\begin{equation}\frac{dy}{dt} = ay + \cos\left(\omega t\right)\label{men-32}\tag{4.4.9}\end{equation}\begin{equation}y=y(0)\:at\:t=0\label{men-33}\tag{4.4.10}\end{equation}

This is the standard growth equation with a sinuosoidal input term \(\cos\left(\omega t\right)\).

We look for a particular solutions of the form:

\begin{equation*}y_p = M\cos\left(\omega t\right) + N\sin\left(\omega t\right)\end{equation*}

where \(M\) and \(N\) are constants.

Now try and solve this using SageMath.

Solution