7. Second Order Homogeneous Linear DEs With Constant Coefficients
The general form of the second order differential equation with constant coefficients is
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):
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:
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
Solve for the current i at time t > 0.
Example 2
Solve the following equation in which s is the displacement of a object at time t.
given that
s = 1,
when t = 0
(That is, the object's position is 1 unit and its velocity is 3 units at the beginning of the motion.)
Example 3
Solve the equation 
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