37 steps, vector t, and two components y, the two columns containing sine and cosine. Now if I go back to the command window, and ask to capture the output in t and y, I then get vectors of output.
And here we get a graph of cosine t starting at 1, and sine t starting at 0. If I call ODE45 with no output arguments, it just plots the solution automatically. 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.
#System of equations solver matlab plus
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.
#System of equations solver matlab how to
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. Many mathematical models involve high order derivatives.
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