Solving second order differential equation with initial conditions

Hi, I am completely new to Matlab and having trouble solving and plotting this second order DE.

I am asked to solve the equation: D^2y/dt + p(dy/dt) + 2y = cos(w*t) with the conditions y(0) = 0 and y'(0) = 0

Assuming p = 0 and w cannot be 0.

This is my attempt at a solution:

syms y(t) p w t

p = 0;

y = dsolve('D2y + p*Dy + 2*y = cos(w*t)','y(0) = 0','Dy(0) = 0','t');

When I enter this into Matlab, I get no answer. Any help on what I could do?

I am also asked to plot solution of the y(t) vs t for other values of w such as 0.5,0.6, etc. How would I go about doing this? Any help or solutions would be greatly appreciated!

Answer 1

[Star Strider]

This may be obsolete syntax:

y = dsolve('D2y + p*Dy + 2*y = cos(w*t)','y(0) = 0','Dy(0) = 0','t');

since I do not know what version of MATLAB you are using.

The code I posted in your previous Question Solve Second Order Differential Equation with Initial Conditions gives you the symbolic answer, and an anonymous function (that I put on one line here):

Yfcn = @(C,t,w) C.*sin(sqrt(2.0).*t)+(cos(sqrt(2.0).*t)-cos(t.*w))./(w.^2-2.0);

You have to supply the ‘C’ constant, but otherwise you can simply use the output from meshgrid to create your ‘t’ and ‘w’ matrices and supply them as arguments to the ‘Yfcn’ function. Then use surf or mesh with the matrix produced by a call to ‘Yfcn’ to plot the result.

[Francisco Zapata]

I am using Matlab R2015b. I'm still having trouble understanding. Could you provide an example of a full working code for me to use as a guide?

[Star Strider]

First, the syntax you used in your dsolve call is obsolete in R2015b (the same version I’m using).

Second, I did in my Answer to your previous Question. I’m re-posting it here:

syms y(t) p w t 
p = 0;
Deq = diff(y,2) + p*diff(y) + 2*y == cos(w*t);
Y = dsolve(Deq, diff(y(0)) == 0, y(0) == 0);
Y = simplify(Y, 'steps', 20);
Yfcn = matlabFunction(Y)

It should run without error in R2015b.

[Francisco Zapata]

Okay I'm understanding but I don't know how to implement the meshgrid to plot the function with the values of w. Do you have code for that as well? Sorry I'm really new and have not used most of these functions.

[Star Strider]

MATLAB can be a bit intimidating at the beginning because it’s possible to do so much with it.

This is what I would do:

Yfcn = @(C,t,w) C.*sin(sqrt(2.0).*t)+(cos(sqrt(2.0).*t)-cos(t.*w))./(w.^2-2.0);
t = linspace(0, 10, 25);
w = linspace(0, 2*pi, 50);

[T,W] = meshgrid(t,w);

C = 1;                                                  % Constant
Y = Yfcn(C,T,W);

figure(1)
surf(T, W, Y)
grid on
xlabel('Time')
ylabel('Angular Frequency (rad)')
zlabel('Y')