Calculus
Separation of variables
Get all the y's on one side, all the x's on the other, then integrate both.
For a separable DE dy/dx = f(x)g(y): get all y's on one side, all x's on the other, and integrate.
Walk through
Step 1 of 5
Check it's separable
dy/dx = xy factors as (x)·(y) — an x-only part times a y-only part. That's the green light for separation of variables.
Treating dy/dx as a fraction (splitting it into dy and dx) is a shortcut that *only* works for separable equations — but here it's completely legitimate.
Try it
Solve dy/dx = ky (exponential growth/decay).
dy/y = k dx → ln|y| = kx + C → y = A e^(kx). With y(0) = y₀: y = y₀ e^(kx). This is the model for populations, radioactive decay, compound interest.
Your turn
Solve dy/dx = x/y with y(0) = 2.
Try it
dy/dx = xy
dy/y = x dx. ln|y| = x²/2 + C. y = A·e^(x²/2).