Linear differential equations

A first-order linear equation has the form dy/dx + Py = Q, where P and Q are functions of x alone (or constants). y and dy/dx appear only to the first power and are not multiplied together.

Multiply both sides by the integrating factor IF = e^(∫P dx). The left side becomes the derivative of y × IF, so y × IF = ∫ Q × IF dx + C.

Worked example: dy/dx + 2y = 4. IF = e^(2x), so y e^(2x) = ∫ 4e^(2x) dx = 2e^(2x) + C and y = 2 + Ce^(−2x). With y(0) = 5, C = 3 and y = 2 + 3e^(−2x), which settles towards 2.

Worked example: dy/dx + y/x = x² for x > 0. IF = e^(log x) = x, so xy = ∫ x³ dx = x⁴/4 + C. Through (1, 1): C = 3/4, giving y = x³/4 + 3/(4x).

Worked example: dy/dx + y tan x = sec x. IF = e^(∫tan x dx) = sec x, so y sec x = ∫ sec²x dx = tan x + C and y = sin x + C cos x.

Some equations are linear in x instead: dx/dy + P₁x = Q₁, with P₁ and Q₁ functions of y. Then IF = e^(∫P₁ dy) and x × IF = ∫ Q₁ × IF dy + C.

Worked example: dx/dy − x/y = y (y > 0). IF = e^(−log y) = 1/y, so x/y = ∫ 1 dy = y + C and x = y² + Cy.

Before finding IF, rewrite the equation so that dy/dx has coefficient 1; P is read from that standard form.