# 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