7. Second Order Homogeneous Linear DEs With Constant Coefficients

The general form of the second order differential equation with constant coefficients is

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where a, b, c are constants with a > 0 and Q(x) is a function of x only.

Homogeneous Equation

In this section, most of our examples are homogeneous 2nd order linear DEs (that is, with Q(x) = 0):

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where a, b, c are constants.

Method of Solution

The equation

am2 + bm + c = 0

is called the Auxiliary Equation (A.E.)

The general solution of the differential equation depends on the solution of the A.E. To find the general solution, we must determine the roots of the A.E. The roots of the A.E. are given by the well-known quadratic formula:

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Summary

Differential Equation:

ay'' + by' + c = 0

Associated auxiliary equation:

am2 + bm + c = 0

Nature of roots Condition General Solution
1. Real and distinct roots,
m1, m2		 b2 − 4ac > 0 y = Aem1x + Bem2x
2. Real and equal roots, m b2 − 4ac = 0 y = emx(A + Bx)

3. Complex roots

m1 = α + jω

m1 = α − jω

 b2 − 4ac < 0 y = eαx(A cos ωx + B sin ωx)

Example 1

The current i flowing through a circuit is given by the equation

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Solve for the current i at time t > 0.


Answer


Example 2

Solve the following equation in which s is the displacement of a object at time t.

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given that

s = 1, $\dfrac{ds}{dt}=3$ when t = 0

(That is, the object's position is 1 unit and its velocity is 3 units at the beginning of the motion.)


Answer


Example 3

Solve the equation MATH


Answer




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