# Linear differential equations of order N

A linear order differential equation no It is of the form:

fno(x) y(n) + fn-1(x) y(n-1) +… + F2(x) y" + f1(x) y '+ f0(x) y = k (x)

where k (x) and the coefficients fi (x) are functions of x.

## Classifications

Linear equation homogeneous (k (x) = 0), or linear equation inhomogeneous (k (x) 0).

Linear equation:

of constant coefficients (f0f1f2,…, Fno constants)
of variable coefficients (at least one fi variable)

## Exact Differential Equations

If P and Q have continuous partial derivatives, then:

P (x, y) dx + Q (x, y) dy = 0

is exact differential equation if and only if

Ex: (3x² - 2y³ + 3) dx + (x³ - 6xy² + 2y) dy = 0

P (x, y) = 3x²y - 2y³ + 3 and Q (x, y) = x³ - 6xy² + 2y and therefore Px = Qx and the differential equation is exact.

## Theorem

The first order linear differential equation y '+ P (x) y = Q (x) can be transformed into a differential equation of separable variables by multiplying both members by the integral factor .

Ex: Solution: The equation has the form of the theorem where, P (x) = -3x² and Q (x) = x²

By the theorem: Multiplying all terms by the integral factor:   - 3x² y = x² or =  dx =  + C

The multiplication by gives the solution: Next Contents: Logarithmic and Exponential Functions