Examples of differential equations

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Differential equations arise in many problems in physics, engineering, etc. The following examples show how to solve differential equations in a few simple cases when an exact solution exists.

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Separable first-order linear ordinary differential equations

A separable linear ordinary differential equation of the first order has the general form:

where f(t) is some known function. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side),

Integrating, we find

where C is a constant. Then, by exponentiation, we obtain

with A another arbitrary constant. It is easy to confirm that this is a solution by plugging it into the original differential equation:

Some elaboration is needed because ƒ(t) is not necessarily a constant—indeed, it might not even be integrable. Arguably, one must also assume something about the domains of the functions involved before the equation is fully defined. Are we talking about complex functions, or just real, for example? The usual textbook approach is to discuss forming the equations well before considering how to solve them.

Non-separable first-order linear ordinary differential equations

Some first-order linear ODEs (ordinary differential equations) are not separable as in the above example. In order to solve non-separable first-order linear ODEs one must use what is known as an integrating factor. Consider first-order linear ODEs of the general form:

The method for solving this equation relies on a special integrating factor, μ:

We choose this integrating factor because it has the special property that its derivative is itself times the function we are integrating, that is:

Multiply both sides of the original differential equation by μ to get:

Because of the special μ we picked, we may substitute /dx for μ p(x), simplifying the equation to:

Using the product rule in reverse, we get:

Integrating both sides:

Finally, to solve for y we divide both sides by μ:

Since μ is a function of x, we cannot simplify any further directly.

A simple example

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension/compression of the spring. For now, we may ignore any other forces (gravity, friction, etc). We shall write the extension of the spring at a time t as x(t). Now, using Newton's second law we can write (using convenient units):