7. Second Order Homogeneous Linear DEs With Constant Coefficients
The general form of the second order differential equation with constant coefficients is
`a(d^2y)/(dx^2)+b(dy)/(dx)+cy=Q(x)`
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):
`a(d^2y)/(dx^2)+b(dy)/(dx)+cy=0`
where a, b, c are constants.
Method of Solution
The equation
`am^2 + 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:
`m=(-b+-sqrt(b^2-4ac))/(2a)`
Summary
Differential Equation:
ay'' + by' + cy = 0
Associated auxiliary equation:
am^{2} + bm + c = 0
Nature of roots | Condition | General Solution |
---|---|---|
1. Real and distinct roots, m_{1}, m_{2} |
b^{2} − 4ac > 0 | `y = Ae^(m_1x)+Be^(m_2x)` |
2. Real and equal roots, m | b^{2} − 4ac = 0 | `y = e^(mx)(A + Bx)` |
3. Complex roots m_{1} = α + jω m_{2} = α − jω |
b^{2} − 4ac < 0 | `y = e^(alphax)(A cos \omega x + B sin \omega x)` |
Example 1
The current i flowing through a circuit is given by the equation
`(d^2i)/(dt^2)+60(di)/(dt)+500i=0`
Solve for the current i at time t > 0.
Here's a typical solution graph for Example 1 with arbitrary values `A=0.1` and `B=2`:
Graph of a possible current at time `t`, given by `i=0.1e^(-10t)+2e^(-50t)`.
Example 2
Solve the following equation in which s is the displacement of an object at time t.
`(d^2s)/(dt^2)-4(ds)/(dt)+4s=0`
given that
`s = 1`, `(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.)
Example 3
Solve the equation
`(d^2y)/(dx^2)-2(dy)/(dx)+4y=0`
Here's a typical solution graph for Example 3 with arbitrary values `A=0.5` and `B=0.2`:
Graph of a typical solution given by `y=e^x(0.5 cos sqrt3 x+0.2 sin sqrt3 x)`.
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