Differential equation:

an equation relating a variable, a function, and the derivatives of that function

Problem: to find the function

General solutions – usually have “constants of integration”

Boundary value problems or initial value problems give additional information that allow us to find a particular solution.

Example:  Swinging pendulum

L = length

g = acceleration due to gravity

q = angle of the pendulum away from vertical

ignoring friction

Motion of the pendulum is described by .

For an initial value problem, we might also be given .

This equation is hard (impossible) to solve.  There are two approaches:

1.             simplify the equation. 

for small values of q, sin q ~ q, so, if we assume that the pendulum doesn’t swing very high, we can make the simpler equation

Solve this by standard means.  Assume q = e^lt, so q” = l²e^lt.  

Substitute these into the DE and get

  .  

Divide by the exponential, solve for l, and get two solutions, two constants of integration, and use your initial values to find them.  

This is what you learned (or will learn) in DE’s.

2.             Approximate, using numerical methods.

This is a Second Order equation, because it involves second derivatives.  We will start with First Order equations, that involve only first derivatives.

First order equations

  The first is explicit, the second is implicit.

Slope fields – implicit vs explicit

How to tell if your slope field messes up: Lipschitz conditions