Princeton Review AP Calculus BC, Chapter 9: Differential Equations

In related rates we saw how we can model the change in one thing related to another with derivatives, and differential equations are similar!
* Oftentimes, variables are not constant, so we have to represent their change using a derivative (Ex. the change in y is dy/dx)
* A differential equation models the change in one variable with respect to another
Slope fields show us what the slopes look like at points on a graph

This is for the equation dy/dx = x
  * Remember that this is a derivative so it will show us the slope at these points!
  * All you have to do to construct a slope field is plug in your x/y (or both) values into your differential equation and draw that as your slope
    * Ex. The slope at x = -1 would be -1 (because dy/dx = x)

dy/dx = x solution

The AP exam might also require you to sketch a solution curve given a slope field!

All you have to do is “flow” with the slopes
* Make sure you don’t cross abruptly or draw a line that doesn’t follow the slope
* Because this is by hand, it doesn’t have to be exact, just try and go with the tangent lines!

If you’re given a differential equation where the derivative of a function is equal to some other function, you have to solve for the original! You can do this by taking the integral (antiderivative) of both sides!
A good memory trick is that differential equation problems will be SIPPY problems
  * S: separate (dy and dx on separate sides)
  * I: integrate (remove the derivative)
  * P: Plus C (add your c value to your integral)
  * P: Plug in your initial condition
  * Y: Y equals (solve to find what y is)
Example: If dy/dx = 4x/y and y(0) = 5 we need to solve for y
  * Start by separating → ydy = 4xdx
  * Then integrate → ∫ydy = ∫4xdx → y^2/2 = 2x^2 + C
    * (Make sure you add C!)
  * Plug in → (5)^2/2 = 2(0)^2 + C
    * C = 25/2
  * Now set y equals → y = 2x^2 + 25/2