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The ODE consists of equations involving functions and their derivatives 常微分方程.

We use 斜率fields to illustrate the concept of our 微分equations. We also call slopefields fields direction.

Use the function in MATLAB slope_field()to plot the slope field of a first-order general 微分equation

slope_field()Functions take three parameters. The first parameter is fthe function we are dealing with 带有 x 和 y 参数的方程.

The second parameter is the minimum and maximum limits that our xparameter lies within. The third parameter is the minimum and maximum limits that our yparameter lies within.

These limits are often called xthe and ydomains. The function slope_field()helps us plot the slope field of our equation, while returning a graphics handle for our field.

Assume that our difference equation is:

$$
\frac {dy} {dx} = \frac {3y} {1-2x}
$$

We set the domain of x to be [-1,12]and the domain of y to be [-4, 5].

This means that our function f(x,y) = 3y/(1-2x)is

f = @(x,y) 3*y/(1-2*x);
figure;
slope_field(f,[-1,12],[-4,5]);
xlabel('$x$','interpreter','latex','fontsize',17);
ylabel('$y$','interpreter','latex','fontsize',17);
title('Slope Field for $\displaystyle\frac{dy}{dx}=\frac{3y}{1-2x}$',...
    'interpreter','latex','fontsize',17);

Output:

In this example, we used slope_field()the function with the default settings and visualized 微分the slope field for the equation we wanted.

Use the function in MATLAB quiver()to plot the slope field of a first-order general 微分equation

The function quiver()takes four parameters:

• X coordinate
• Y coordinate
• The directional component of the X coordinate represented by U.
• The directional component of the Y coordinate represented by V.

This function returns a graphical representation of the slope field in the form of arrows with coordinates X 和 Yand direction components U 和 V.

Assume that our differential equation is:

$$
\frac {dx} {dt} = x^5+6xy-3y
$$

$$
\frac {dy} {dt} = -8x+sin\left(2yx\right)
$$

[x,y] = meshgrid(-3:0.1:3);
dx = x.^5+6*x.*y-3*y;
dy = -8*x+sin(2*x.*y);
r = ( dx.^2 + dy.^2 ).^0.5;
px = dx./r;
py = dy./r;
quiver(x,y,px,py);

Output: