In details

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:

- 3x²y = x² or = dx = + C

The multiplication by gives the solution:

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