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

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Jordan Stanley
Jordan Stanley 2022-4-11
评论： Jordan Stanley 2022-4-11
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!

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Sam Chak
Sam Chak 2022-4-11
编辑：Sam Chak 2022-4-11
@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')
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Sam Chak
Sam Chak 2022-4-11
编辑：Sam Chak 2022-4-11
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.

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