How To Plot Directional Field of 2nd Order Differential Equation IVP

Hello,

I have the second order differential equation initial value problem, y'' + 2y' + y = 0, y(-1) = 0, y'(0) = 0.

In MATLAB, I need to plot the directional field of the solution to the equation without the initial conditions.

I have used the meshgrid() command so far and know that I have to use the quiver() command but I don't know how to enter what I need as parameters to plot the solution.

Here is what I have so far...

% Finds solution to the DE
syms y(x) 
Dy = diff(y); 
D2y = diff(y,2); 
ode = D2y + 2*Dy + y == 0; 
ySol = dsolve(ode) 
 
% Sets up directional field 
[x,y]=meshgrid(-3:0.3:3,-2:0.3:2); 
quiver() %Not sure what to include here.

Any help is appreciated!

Answer 1

[Sam Chak]

@Jordan Stanley

Basically, it should look something like this:

[X, Y] = meshgrid(-3:6/14:3, -3:6/14:3);
U = Y;              % x1' = y'
V = - 2*Y - X;      % x2' = y'' = - 2*y' - y
quiver(X, Y, U, V)
% quiver(X, Y, U, V, 1.5)   % can adjust arrow size
xlabel('x')
ylabel('y')

[Jordan Stanley]

Hello and thank you for replying!

What did setting U to equal Y do exactly. Does U equal the y' part of the equation?

I just want to understand what you did here.

[Sam Chak]

Hi @Jordan Stanley

Well, the ODE can be rewritten in the form of a state-space representation.

Begin by defining

Taking the time derivative yields

.

Rewritting the dynamics in system states

.

Back to quiver function, this quiver(X, Y, U, V) command plots arrows with directional components U and V at the Cartesian coordinates specified by the grid of X and Y values.

The directional components U and V mean the motion of the the point that extends horizontally according to U vector, and extends vertically according to V vector. Naturally, these imply the first-order equations. So, if we assign

then

.

Hope this explanation is helpful for you to plot the direction field of the desired ODEs.

[Jordan Stanley]

Okay, I think I understand. Thank you for the help.