This page introduces second order differential equations, focusing on their general form and on homogeneous equations with constant coefficients. These equations appear frequently in physics, engineering, and applied mathematics.
General Form
A second order linear differential equation can be written as:
\[ \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = R(x) \]If \( R(x) \neq 0 \), the equation is called nonhomogeneous.
If \( R(x) = 0 \), the equation becomes:
\[ \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0 \]and is called a second order linear homogeneous differential equation.
General Solution of the Homogeneous Equation
If \( y_1(x) \) and \( y_2(x) \) are two linearly independent solutions of:
\[ \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = 0 \]then the general solution is:
where \( A \) and \( B \) are arbitrary constants.
Note: Two functions are linearly independent if neither is a constant multiple of the other.
Second Order Equations with Constant Coefficients
Homogeneous equations with constant coefficients have the form:
\[ \frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0 \]where \( b \) and \( c \) are constants.
We seek solutions of the form \( y = e^{kx} \). Then:
\[ \frac{dy}{dx} = ke^{kx}, \qquad \frac{d^2y}{dx^2} = k^2e^{kx} \]Substituting into the differential equation gives:
\[ k^2e^{kx} + bke^{kx} + ce^{kx} = 0 \]Factoring out \( e^{kx} \):
\[ e^{kx}(k^2 + bk + c) = 0 \]Since \( e^{kx} \neq 0 \), we obtain the characteristic (auxiliary) equation:
The roots are given by the quadratic formula:
\[ k_{1,2} = \frac{-b \pm \sqrt{b^2 - 4c}}{2} \]Let \( D = b^2 - 4c \) be the discriminant. Three cases arise:
- \( D > 0 \): two distinct real roots
- \( D = 0 \): one repeated real root
- \( D < 0 \): complex conjugate roots
Each case leads to a different form of the general solution.
Worked Examples & References
Explore step-by-step worked examples for each case: