# MATLAB Demo on 7.7-7.8: Differential Equations

## Contents

## Direction Fields

Let's start with a differential equation, say

At each point in the (x,y) plane (at least if y is not zero), this specifies what the slope of any solution curve must be. We can plot these to get a visual impression of the equation.

[xx, yy] = meshgrid(-2:.4:2,0.01:.4:4.01); quiver(xx,yy,xx./xx,2*tanh(.5*xx./yy))

## Solutions

We can solve for the ``general solution'' of the equation.

syms x y gensol=dsolve('Dy=x/y','x') yfun = @(x, C1) eval(vectorize(gensol(1))) hold on for c=0:10 ezplot(yfun(x,c),[-2,2]), axis([-2,2,0,4]) title('some solution curves of dy/dx = x/y') end hold off

gensol = (x^2+C1)^(1/2) -(x^2+C1)^(1/2) yfun = @(x,C1)eval(vectorize(gensol(1)))

## Another Example

Let's look now at a linear equation,

Again we plot the direction field and some solution curves.

[xx, yy] = meshgrid(-pi:.2*pi:pi,-4:4); quiver(xx,yy,xx./xx,2*tanh(.5*cos(xx).*(1-yy))) lingensol=dsolve('Dy=(1-y)*cos(x)','x') yfun2 = @(x, C1) eval(vectorize(lingensol)) hold on for c=-4:4 ezplot(yfun2(x,c),[-pi,pi]), axis([-pi,pi,-4,4]) title('some solution curves of dy/dx + y*cos(x) = cos(x)') end hold off

lingensol = 1+exp(-sin(x))*C1 yfun2 = @(x,C1)eval(vectorize(lingensol))