# Second Order Differential Equations - Generalities

## Generalities

The general form of the second order linear differential equation is as follows
 $\dfrac{d^2y}{dx^2} + P(x) \dfrac{dy}{dx} + Q(x) y = R(x)$
If $$R(x)$$ is not equal to zero, the above equation is said to be inhomogeneous.
If $$R(x) = 0$$, the above equation becomes
$\dfrac{d^2y}{dx^2} + P(x) \dfrac{dy}{dx} + Q(x) y = 0$
and is called second order linear homogeneous differential equation.

## Theorem

If $$y_1(x)$$ and $$y_2(x)$$ are two linearly independent solutions of the homogeneous differential equation $$\dfrac{d^2y}{dx^2} + P(x) \dfrac{dy}{dx} + Q(x) y = 0$$, then the general solution of the above equation may be written as
$y(x) = A y_1(x) + B y_2(x)$
where A and B are constants.
NOTE: Functions $$y_1(x)$$ and $$y_2(x)$$ 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
$\dfrac{d^2y}{dx^2} + b \dfrac{dy}{dx} + c y = 0 \qquad (I)$
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 $$\dfrac{dy}{dx} = k e^{kx}$$ and $$\dfrac{d^2 y}{dx^2} = k^2 e^{kx}$$.
Substitute $$y$$, $$dy/dx$$ and $$d^2 y / dx^2$$ into the differential equation (I) to obtain the equation
$\displaystyle k^2 e^{kx} + b k e^{kx} + c e^{kx} = 0$
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 $$k_1$$ and $$k_2$$ of the auxiliary equation, which is a quadratic equation in k, are given by
$$k_1 = \dfrac{ - b + \sqrt{D} } { 2 }$$ and $$k_2 = \dfrac{ - b - \sqrt{D} } { 2 }$$
where $$D = b^2 - 4c$$.
Since D may be negative, positive or equal to zero, solutions $$k_1$$ and $$k_2$$ may real and distinct when $$D > 0$$, real and equal when $$D = 0$$ and complex conjugate when $$D \lt 0$$. 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