A major key to success in solving differential equations involves classifying the type of differential equation you have. Correct identification will narrow down the list of available techniques that you need to choose from. Specific techniques work only on specific types of differential equations. So you need to start by looking carefully at the type of equation you have. Here is a list of types.
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Order indicates the highest derivative appearing in the differential equation. For example, \( y' - y = 0 ~\) is a first order differential equation because the highest derivative is a first derivative. Similarly, \( y'' - y = 0 \) is a second order differential equation because the highest derivative is a second derivative.
A differential equation of the form \(P(t) y'' + Q(t)y' + R(t)y = G(t) \) is linear since \(P(t)\), \(Q(t)\) and \(R(t)\) are functions of t only, i.e. they do not contain any y's or derivatives of y. Analyzing nonlinear equations is relatively difficult, so it is unlikely you will encounter them in a first semester differential equations course, except under very special circumstances.
Homogeneity (Homogeneous or Inhomogeneous)
There are several meanings of the term 'homogeneous' used in differential equations. For this definition, assume you have a differential equation of the form \( p(t)y'' + q(t)y' + r(t)y = g(t) \).
Meaning 1 - - If \( g(t) = 0 \) in the above form, then the differential equation is said to be homogeneous. The idea is to get all the terms containing y or a derivative of y to one side of the equal sign and all other terms to the other side. If there are no terms without a y or it's derivatives, then \(g(t)\) will be zero and the equation will be homogeneous. If \(g(t) \neq 0 \), then the differential equation is said to be inhomogeneous (or nonhomogeneous).
Meaning 2 - - If all of the expressions \(p(t)\), \(q(t)\), \(r(t)\) and \(g(t)\) can be written as functions of \(y/t\), then it is said to be homogeneous. In this case, we use the technique of substitution to solve this type of differential equation.
Now that you have an idea of how to classify differential equations, you are ready to begin studying specific topics. The next natural topic is slope fields. However, many instructors will go straight to separation of variables. In any case, take some time to enjoy studying differential equations.
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