Linear First and Second Order Differential Equation

Linear First and Second Order Differential Equation

Linear First Oder Differential Equation

The equation in the form dy/dx + Py = Q where P and Q are functions of x only is called Linear Differential Equations since y and its derivatives are of the first degree.

The solution for dy/dx + Py = Q is obtained by multiplying throughout by an Integrating factor Linear First and Second Differential Equation  to become Linear First and Second Differential Equation

Example: Solve the equation dy +4xy dx = 2xdx

Solution:

Rearranging:  dy/dx + 4xy = 2x

then P = 4x and Q = 2x

 

Linear Second Order Differential Equation

Equation in the form a (d^2y/dx^2) + b(dy/dx) + cy = 0 where a, b, and c are constants, is called a linear second order differential equation with constant coefficient

Setting D = d/dx and D^2 = d^2/dx. The following procedures may be followed.

  1. write the equation in D – operator form (aD^2 + bD + C) y=0, substitute m for D and solve the auxiliary equation am^2 + bm + c=0 for m

A. If the roots are real and different (b^2 > 4ac) say Linear First and Second Differential Equation.

Then the general solution is Linear First and Second Differential Equation

B. If the roots are real and equal Linear First and Second Differential Equation twice the general solution is Linear First and Second Differential Equation

C. If the roots are imaginary (b^2 – 4ac) Say Linear First and Second Differential Equation

the general solution is Linear First and Second Differential Equation

Example:  Solve the equation 2(d^2y/dx^2) + 5(dy/dx) – 3y = 0

Solution:

Writing D – operator form: (2D^2  + 5D – 3) y = 0

Substituting m for D gives the auxiliary equation 2m^2 + 5m – 3 = 0 which can be factored as (2m – 1) (m + 3) and the roots are m = ½ and m = -3

Since the roots are real and different the general solution is  Linear First and Second Differential Equation

then the general solution is Linear First and Second Differential Equation

Linear First and Second Differential Equation

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