Question

In Matlap, Please do Matlap code

Consider the second order differential equation y” + 4y = 0, y(0) = 1, y(0) = 1 1. Use ode45 to solve this equation for y over the time interval [0,20] and store the result as y. Warning: When you run ode45, you will get both y and y' in a single matrix; you will have to select the appropriate part to isolate y. 2. Calculate the amplitude of y. Store the result as ymax 3. The template gives an example of how to use Matlab's findpeaks function to estimate the period of a signal. Modify this code to estimate the period of y and store the result as yperiod.

Answer 1

• Inorder to solve a 2nd order ODE with a matlab's ode solver, we have to convert it to state space form first.
• The state space form can be obtained by passing the ODE equatino to odeToVectorField() funciton.
• Then, create a matlab fucntion out of this state space and pass it to ode45.
• Ode45 return a vector of x values and a 2xn vector y of y and y' values.
• the first row of y will be reqiured y values.
• the amplitude of the signal can be found out by using max() funcntion on y(1,:)
• the time period can be found by calculating the x distance between the two consectutive value od result obtained from findpeaks() funciton.

program:

syms y(t)

ss = odeToVectorField(diff(y,2)+4*y==0);

f = matlabFunction(ss,'vars',{'t','Y'});

[x, yv] = ode45(f,[0 20],[1 1]);

y = yv(:,1)

ymax = max(y(11:end))

[peaks,locs] = findpeaks(y);

yperiod = x(locs(4))-x(locs(3))

outputs:

sol = ode45 (f, [020],[1 1]) sol = struct with fields: solver: 'ode45' extdata: [lxl struct] x: [1x53 double] y: [2x53 double] stats: [1xl struct] idata: [1xl struct] ymax = max(sol.y(1,:)) ymax = 1.1072 [peaks, locs] findpeaks (sol.y(1,:)) peaks = 1x7 1.1072 1.0766 1.0524 1.0476 1.0466 ... locs = 1x7 11 19 27 35 43 51 yperiod = sol.x(locs (2))-sol.x(locs (1)) yperiod = 3.2078