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

f_{no}(x) y^{(n)} + f_{n-1}(x) y^{(n-1)} +… + F_{2}(x) y^{"} + f_{1}(x) y '+ f_{0}(x) y = k (x)

where k (x) and the coefficients f_{i }(x) are functions of x.

## Classifications

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

**Linear equation: **

of constant coefficients (f_{0}f_{1}f_{2},…, F_{no} constants)

of variable coefficients (at least one f_{i }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:

- 3x²y = x² or = x²dx = + C

The multiplication by gives the solution:

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