Linear equations

A linear differential equation of first order has the form:

dy/dx + Py = Q, where P and Q contain only x (or are constants).

'Linear' means y and dy/dx appear only to the first power, and are never multiplied together.

Method:

• Rewrite the equation in exactly this form: divide so that dy/dx stands alone with coefficient 1. Read off P and Q, with their signs.
• Find the integrating factor: I.F. = e^(∫P dx).
• Write the solution: y × I.F. = ∫(Q × I.F.) dx + C.

Why it works: multiplying by the I.F. turns the left side into the derivative of (y × I.F.), which can then be integrated directly.

Example: dy/dx + y = e^(–x). P = 1, so I.F. = eˣ. Then y eˣ = ∫e^(–x) eˣ dx = ∫1 dx = x + C, so y = (x + C)e^(–x).

Simplify the I.F. with e^(log u) = u: e^(2 log x) = x², and e^(–log x) = 1/x. e^(∫cot x dx) = e^(log sin x) = sin x.

If the equation is linear in x instead, dx/dy + P₁x = Q₁ with P₁, Q₁ in y only, then I.F. = e^(∫P₁ dy) and x × I.F. = ∫(Q₁ × I.F.) dy + C. Same steps, letters swapped.