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