# Second Order Differential Equations - Generalities

## Generalities

The general form of the second order linear differential equation is as follows

**d**

^{2}y / dx^{2}+ P(x) dy / dx + Q(x) y = R(x)If R(x) is not equal to zero, the above equation is said to be

**inhomogeneous**.

If R(x) = 0, the above equation becomes

**d**

^{2}y / dx^{2}+ P(x) dy / dx + Q(x) y = 0and is called second order linear

**homogeneous**differential equation.

## Theorem

If y1(x) and y2(x) are two linearly independent solutions of the homogeneous differential equation d^{2}y / dx^{2} + P(x) dy / dx + Q(x) y = 0, then the general solution of the above equation may be written as

**y(x) = A y1(x) + B y2(x)**

where A and B are constants.

NOTE: Functions y1(x) and y2(x) are linearly independent if one is not a multiple of the other.

## Second Order Differential Equations With Constant Coefficients

Homogeneous second order differential equations with constant coefficients have the form

**d**

^{2}y / dx^{2}+ b dy / dx + c y = 0where b and c are constants.

Because of the presence of the first and second derivatives in the above equation, solutions of the form y = e

^{kx}are appropriate for the above equation.

If y = e

^{kx}, then dy / dx = k e

^{kx}and d

^{2}y / dx

^{2}= k

^{2}e

^{kx}.

Substitute y, dy/dx and d

^{2}y / dx

^{2}into the differential equation to obtain

k

^{2}e

^{kx}+ b k e

^{kx}+ c e

^{kx}= 0

Factor e

^{kx}out

e

^{kx}(k

^{2}+ b k + c ) = 0

and since e

^{kx}cannot be zero leads to

**k**

^{2}+ b k + c = 0The above equation in k is called the auxiliary equation for the given homogeneous equation. The solutions k1 and k2 of the auxiliary equation, which is a quadratic equation in k, are given by

k1 = [ - b + √D ] / 2 and k1 = [ - b - √D ] / 2

where D = b

^{2}- 4c.

Since D may be negative, positive or equal to zero, solutions k1 and k2 may real and distinct when D > 0, real and equal when D = 0 and complex conjugate when D < 0. All these cases will be discussed in the following pages:

Solve Second Order Differential Equations - part 1

Solve Second Order Differential Equations - part 2

Solve Second Order Differential Equations - part 3