Separation of variables is a method for solving a differential equation. I'll illustrate with some examples.
Example. Solve .
"Solve" usually means to find y in terms of x. In general, I'll be satisfied if I can eliminate the derivative by integration.
First, I rearrange the equation to get the x's on one side and the y's on the other (separation):
Next, I integrate both sides:
I only need an arbitrary constant on one side of the equation. Finally, I solve for y in terms of x, if possible:
Here's a convenient trick which I'll use in these situations. Think of as . Move the to the other side:
Now define :
The last step makes the equation nicer, and it's easier to solve for the arbitrary constant when you have an initial value problem.
Example. Solve , where .
In this case, solving would produce plus and minus square roots, so I'll leave the equation as is.
Plug in the initial condition: When , :
Hence, the solution is
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