First Order Equations Explained with Simple Animations

First order differential equations are fundamental to understanding how things change.A first order differential equation involves a function and its first derivative.Let's visualize this with a simple function and its rate of change.Here's our function, a simple quadratic curve.The derivative at any point represents the slope of the tangent line at that point.The derivative itself forms another function, showing how the rate of change varies.One practical example is population growth, where the rate of change is proportional to the current population.This creates an exponential growth curve, where faster growth occurs at higher populations.Another example is Newton's Law of Cooling, where the rate of temperature change depends on the temperature difference.The cooling rate is highest when the temperature difference is greatest, leading to this characteristic decay curve.These are just a few examples of first order differential equations in action.Separable equations are differential equations where we can separate the variables x and y.We can rearrange these equations to have all y terms on one side and all x terms on the other.For example, dy dx equals x y can be separated into one over y dy equals x dx.The solutions to separable equations often form families of curves, as shown here.Linear first order equations always follow this standard form, where P of x and Q of x are functions of x only.Here's an example of a linear equation: dy dx plus 2xy equals x squared.Linear equations have distinctive solution curves that never intersect.Exact equations involve two functions M and N that satisfy certain conditions.Here's an example of an exact equation. Notice how both x and y appear in each term.The solutions to exact equations often form level curves of some potential function.Let's compare these three types of first order differential equations.