This method is appropriate if you have an equation of the form dy/dx=f(x)g(y).
For example, dy/dx=y2sin(x) or dy/dx=x3ey.
In this case, we cannot just integrate the right-hand side, because the y is involved there.
Instead we do something else, as demonstrated by this example.
Example 1. Solve:
The idea, as the name of the method suggests, is that we separate the two variables x and y.
Remember that the reason we couldn't use direct integration was because of the y-dependence on the right-hand side. The first step is to rearrange the equation so that there is NO y-term on the right-hand side.
We do this by moving all the parts that involve y onto the left-hand side.In this case that means multiplying both sides by y-1. This gives:
Now the final step is to integrate both sides with respect to x:
Using the function-of-a-function rule on that left-hand integral we can rewrite the equation as:
Now we can just do those integrals, to get what we were after in the first place: a relationship between x and y with no derivatives involved:
Here's some more examples of the same thing, you try them then click to check your answers.
