Variables separable

The simplest equations to solve are variables separable: all the y parts can be moved to one side with dy, and all the x parts to the other side with dx.

If dy/dx = g(x) × h(y), rewrite it as dy/h(y) = g(x) dx, then integrate both sides.

Add one constant C, on one side only. (Constants from both sides would combine into one anyway.)

Example: dy/dx = 2xy. Separate: dy/y = 2x dx. Integrate: log|y| = x² + C. So y = ±e^C e^(x²), which we write as y = Ae^(x²), with A a new constant.

Look for disguised products:

• e^(x + y) = eˣ × eʸ, and e^(x – y) = eˣ ÷ eʸ.
• 1 + x + y + xy = (1 + x)(1 + y).

Integrals that come up again and again:

• ∫dy/y = log|y|
• ∫dy/(1 + y²) = tan⁻¹y
• ∫dy/√(1 – y²) = sin⁻¹y

Tidy the answer with log rules. If you get log|u| + log|v| = C, write C as log A: then log|uv| = log A, so uv = A. This makes the final answer neat and is how most textbook answers look.