I'm going to use ODE45, and if I call it with no output arguments, ODE45 of the differential equation f, t span the time interval, and y0 the initial condition. I need a column vector, 0, 1, for the two components. And I'm going to ask for output in steps of 2 pi over 36, which corresponds to every 10 degrees like the runways at an airport. We're going to integrate from 0 to 2pi, because they're trig functions. Let's bring up the MATLAB command window. When we write the initial condition in the MATLAB, it's the column vector 0, 1. That implies they solution is sine t and cosine t. Or in vector terms, the initial vector is 0, 1. In terms of the vector y, that's y1 of 0, the first component of y is 0. Now for some initial conditions- suppose the initial conditions are that x of 0 is 0, and x prime of 0 is 1. All the content is in the second component, which expresses the differential equation. The first component here is just a matter of notation. First it's a vector now, a column vector. f is an autonomous function of t and y, that doesn't depend upon t. When we write this as an autonomous function for MATLAB, here's the autonomous function. Y2 prime is minus y1 is the actual harmonic oscillator differential equation. That's just saying that the derivative of the first component is the second. The first components says y1 prime is y2. The vector system is now y1, y2 prime is y2 minus y1. Once you've done that, everything else is easy. so y2 prime is playing of x double prime? Do you see how we've just rewritten this differential equation. So the differential equation now becomes y2 prime plus y1 equals zero. Then the derivative of y is the vector with x and x double prime. We're just changing notation to let y have two components, x and x prime. This is a vector with two components, x, and the derivative of x. So to write it as a first order system, we introduced the vector y. Let's see how to do that with a very simple model, the harmonic oscillator. So we have to rewrite the models to just involve first order derivatives. But the MATLAB ODE solvers only work with systems of first order ordinary differential equations. All the tutorials are completely free.Many mathematical models involve high order derivatives. This website contains more than 150 free tutorials! Every tutorial is accompanied by a YouTube video. The inverse Laplace transform can be computed by executing the following code lines We use MATLAB to compute the inverse Laplace transform. Taking into account that and, and by transforming the expression ( 3), we obtainīy applying the inverse Laplace transform to ( 4), we can obtain as function of. By applying the Laplace transform to ( 2), we obtain Let us apply the Laplace transform to equation ( 2). Let us assume that initial conditions are and. We perform the tests using the following differential equation The approach that is used for comparison is based on the Laplace transform. The two approaches should produce results that match. The idea is to compare this approach with another approach for computing the analytical solution. The result is shown in the figure below.įinally, let us verify that this approach produces accurate results. First, we choose the plotting interval, and then similarly to the MATLAB function plot(), we can use the function to plot the solution.
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