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integrate

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I am taking a course in differential equations.

So far we have learned the following topics/methods:

- Separable equations

- Direction fields

- Exact equations

- Linear ODE's - Homogeneous & Non-Homogeneous equations

1. How do you know if an equation is linear or non-linear. What is the quickest way to tell?

>> In order to be linear, does it have to have the form: y' + p(x)y = r(x) (Known as the Standard Form). Where p(x) and r(x) are functions of x only or constants?

2. When y' + p(x)y = 0; where r(x) = 0. Is this a Homogeneous equation? Why?

3. When y' + p(x)y = r(x); where r(x) is a function of x only or a constant. Is this a Non-Homogeneous equation? Why?

4. Finally, can someone explain a methodical way to draw direction fields for a given ODE. It has something to do with isoclines. When you draw the direction fields you get tangents (slope lines) which define the direction of the solution curves y(x). Can someone give me an idea how to do this with an example.

Thanks.