Variables separable

A first-order, first-degree equation dy/dx = F(x, y) is separable if F(x, y) = g(x) h(y), a product of a function of x alone and a function of y alone.

Method: move every y to one side with dy and every x to the other with dx, then integrate both sides: ∫ dy/h(y) = ∫ g(x) dx + C. One constant on one side is enough.

Worked example: dy/dx = x/y. Then y dy = x dx, so y²/2 = x²/2 + c, that is y² − x² = C. Through (2, 1): 1 − 4 = C = −3, giving x² − y² = 3.

Worked example: dy/dx = e^(x − y) = eˣ e^(−y). Then eʸ dy = eˣ dx and eʸ = eˣ + C. With y(0) = 0, C = 0 and the solution is y = x.

Worked example: dy/dx = 2xy with y(0) = 3. Then dy/y = 2x dx, log|y| = x² + c and y = Ae^(x²). The condition gives A = 3, so y = 3e^(x²).

Worked example: dy/dx = y cos x with y(0) = 2 gives log|y| = sin x + c, so y = 2e^(sin x).

Dividing by h(y) assumes h(y) ≠ 0. Values where h(y) = 0 (such as y = 0 for dy/dx = 2xy) give constant solutions that should be checked separately.