Dear Matlab, My Professor showed us in class today how to use ode45 to solve a differential equation numerically. I would like to use it on my homework but I have a second order differential equation and the example I saw in class was only first order. Can ode45 be used for second order diffeqs?

Yes, ode45 function can be used for solving 2nd order differential equations.

Explanation:

The ode45 function can be used to solve 2nd order differential equations if we convert the 2nd order differential equation into a system of first order differential equations.

Suppose we have the following 2nd order differential equation and we want to plot its solution for 0 ≤ t ≤ 10 with initial conditions y(0) = 2, and y'(0) = 1

d²y/dt² + 6dy/dt + 4y = -sin(3t)

make d²y/dt² subject of the equation

d²y/dt² = - 6dy/dt - 4y - sin(3t)

define variable x such that

dy/dt = x    eq. 1

Now the above 2nd order differential equation can be written in terms of x as

dx/dt = -6x - 4y + sin(3t)  eq. 2

in vector form

z = [x; y]

dz = [dx/dt; dy/dt]

where z and dz are column vectors

So we have successfully converted the 2nd order differential equation into two 1st order differential equations.

Now we have to translate it into a Matlab function.

Matlab Function:

function dz = func(t, z)

dz=[-6*z(1) -4*z(2) -sin(3*t); z(1)];

end

Where z(1) means x and z(2) means y

Save this function as func.m file in the Matlab directory.

Now we need driver code to call this function and plot its solution

Driver Code:

initial = [2; 1];  % given initial conditions of differential equation

range = [0 10]; % plot the solution in the range 0 ≤ t ≤ 10

[t, z]=ode45(@func,range,initial);   % call 'func' in the ode45 function

plot(t, z(:,2))   % plot the results

grid on

Output:

Please refer to the attached image

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