## 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 = 0 and 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 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 = 0 where 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 = 0
The 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 |