Previous page : A few examples from the kinematics

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Say we have an equation like

This equation tells us what the slope of a solution will be at any given point. If we for example are at (4, 2) then the slope would be 8. A graph of the slopes are given below.

Such a graph is called a slope field.

Since each point has one and only one slope, a solution through it would correspond to one and only one curve. But how to find the solutions? We will do a little trick here and separate the two parts of the differential to get

Next we integrate both sides to get

or

We will get a constant of integration from each integral, but they can be combined to one. The above gives us

But since the sign can be absorbed by the new constant we finally get

A graph of a few of these solutions would look something like this.

To check that this really is the solution we can differentiate what we have to get

But would this trick work in general?

Say we have an equation of the form

From this we get

or

and thus

as a solution. Differentiating this with respect to x gives us

using the chain rule. This gives

I.e. what we started with. We can thus solve equations of the form

by separating the equation to an *y* and *x* part. An equation of the form

can be separated too and the proof lies in the fact that multiplying with something is to divide by its reciprocal, so the above can be rewritten in the former form.

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