What is the 'separation of variables' method and when can I use it?
You can use it whenever the equation factors as dy/dx = g(x)·h(y) — the x-stuff and y-stuff multiply separately.
You can use it whenever the equation factors as dy/dx = g(x)·h(y) — the x-stuff and y-stuff multiply separately. Rearrange to put every y (with dy) on one side and every x (with dx) on the other: dy/h(y) = g(x) dx. Then integrate both sides independently and add a single constant. Applying an initial condition then pins down that constant to give the particular solution. It's the first method to try on a first-order equation.
Key point
If dy/dx = g(x)h(y), separate to dy/h(y) = g(x)dx and integrate both sides.
Common mistake
integrating before getting all the y's and x's onto opposite sides.
Memory tip
separate (y with dy, x with dx), then integrate each side.
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