Direct Integration

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The Method of Direct Integration

The simplest differential equations to solve are those in the form of . These differential equations can be solved by directly integrating both sides to get that for any antiderivative and for as a constant we have that:

For example, suppose that we wanted to solve the differential equation . Then we have that for as a constant:

Another type of differential equation that can be solved by direct substitution are differential equation for which and are constants, in the following form:

For and we can rewrite the differential equation above as:

Notice that if then and so we have:

Generally, since is an arbitrary constant, we have that is also an arbitrary constant which we will denote as , and hence, the solutions to are given by . Notice that we must proceed with some caution, since we must check if yields a solution. In this case, since if then which was omitted earlier. In most cases, will be a solution to the differential equation.

Let's look at an example of using the method if direction integration described above.

Example 1

Find all solutions to the differential equation .

We will start by factoring the righthand side of this equation to get . We now divide both sides of this equation by , and so for we have rewritten the above differential equation as:

Now we can rewrite our differential equation above as follows:

We have now reduced our original differential equation to the first type of differential equation we mentioned at the beginning of this page. We will now integrate both sides with respect to and apply the Fundamental Theorem of Calculus to get:

Cleaning up our solution, we get that , for as a constant (noting that if then is a solution to .

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