First order differential equations have the form: \( F(x,y,y') \), which means it's the relationship between the independent variable \( x \), the unknown function \( y=f(x) \), and its first derivative.
Examples:
\( \frac{dE}{dt} = 0.2E \quad \text{with} \quad E(t = 0) = 200 \)
This is a first-order differential equation describing exponential growth, where the rate of change of \( E \) is proportional to \( E \) itself, with an initial condition at \( t = 0 \).
Differential Equations (DE) are equations that establish a relationship between the independent variable \( x \), the unknown function \( y=f(x) \), and its derivatives \( y', y'', y''' \), etc.
If the unknown function depends on only one variable, it's called an Ordinary Differential Equation. If it depends on more than one variable, it's called a Partial Differential Equation.
In DEs, we find a general solution that geometrically represents a family of curves in the coordinate plane, called Integral Curves. The particular solution represents one curve from this family.