Differential Equations
Study of equations involving derivatives and their applications.
Advanced Topics
Second-Order Differential Equations
Stepping Up: Second Derivatives
These equations involve the second derivative, like \( y'' \) or \( \frac{d^2y}{dx^2} \).
General Form
\( a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x) \)
Here, \( a \), \( b \), and \( c \) are constants.
Homogeneous vs. Nonhomogeneous
- Homogeneous: \( f(x) = 0 \)
- Nonhomogeneous: \( f(x) \neq 0 \)
Applications
These equations describe systems with acceleration, like springs or circuits.
Key Formula
\[a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)\]
Examples
A mass on a spring: \( m\frac{d^2x}{dt^2} + kx = 0 \)
An electrical circuit with a resistor, capacitor, and inductor.
In a Nutshell
Second-order equations handle problems involving acceleration or curvature.