Differential Equations
What is a Differential Equation?
A differential equation is simply an equation that contains a derivative (like \(\dot{x}\) or \(\frac{dy}{dx}\)).
- In algebra (like \(x + 5 = 10\)), you solve for a number.
- In differential equations, you solve for an entire function.
The equation itself is a rule that describes how a function changes. Our goal is to find the original function \(x(t)\) that follows that rule.
Categories of Differential Equations
These equations describe different kinds of systems. The rule can depend on the function’s value (\(x\)), its input (\(t\)), or other derivatives.
1. Rate Depends on Current Value (Proportional Growth)
This is a very common category. \(\dot{x} = ax\) This rule describes any system where the rate of change (\(\dot{x}\)) is directly proportional (by a scaling factor \(a\)) to the current value (\(x\)).
- Example: A bank account, where the rate you earn interest (\(\dot{x}\)) is proportional to the money you currently have (\(x\)).
- Solution: This leads to the exponential function \(x(t) = K e^{at}\).
- Read the full derivation here
(We will discover many other types, including second-order equations that describe oscillations.)