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^{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 = 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 are given by
k1 = [  b + sqrt(D) ] / 2 and k1 = [  b  sqrt(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
