Chapter Contents

• Differential Equations
• Predicting AIDS - a DEs example
• 1. Solving Differential Equations
• 2. Separation of Variables
• 3. Integrable Combinations
• 4. Linear DEs of Order 1
• 5. Application: RL Circuits
• 6. Application: RC Circuits
• 7. Second Order DEs - Homogeneous
• 8. Second Order DEs - Damping - RLC
• 9. Second Order DEs - Forced Response
• 10. Second Order DEs - Solve Using SNB
• Differential Equations Problem Solver

• Comments, Questions?

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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:

am² + bm + c = 0

Nature of roots	Condition	General Solution
1. Real and distinct roots, m1, m2	b2 − 4ac > 0	`y = Ae^(m_1x)+Be^(m_2x)`
2. Real and equal roots, m	b2 − 4ac = 0	`y = e^(mx)(A + Bx)`
3. Complex roots m1 = α + jω m2 = α − jω	b2 − 4ac < 0	`y = e^(alphax)(A\ cos\ ωx + B\ sin\ ωx)`

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.

Answer

Example 2

Solve the following equation in which s is the displacement of a 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.)

Answer

Example 3

Solve the equation

`(d^2y)/(dx^2)-2(dy)/(dx)+4y=0`

Answer