6. Ordinary Differential Equations

what are DEs

equation with an unknown function (y) and its derivative (y’, y’’)

what the difference between an ODE and a DE

OD whose unknown function (y) depends on only 1 independent variable (x)

are antiderivatives DE’s?

yes

option meaning

contract that lets the buyer to buy/sell a stock at a specific price

independent variable and how do you memorize it

Input variable

dependent variable and how do you memorize it

value depends on the independent variable

order of a DE

order of the highest derivative of the unknown function (y’, y’’)

notation of 2nd order derivative using d

(d²f)/(df²)

solution of a DE

any function y=y(t) which itself and its derivatives satisfy the DE

general vs particular solution of a DE

General Solution: A family of functions where

• every function in the family is a solution of the DE

• every solution of the DE is in the family

Particular Solution: individual functions in the family

whats an IVP

initial value problem where theres some condition

form of a separable DE

dy/dx = f(x) * g(x)

how to find the unknown function, whose 1st derivative is the known function

take antiderivative of known function

6 steps to solve separable DE

1. separate dy and dx by multipling by dx on both sides

2. move all x to one side and all y to other side

3. take integral of both sides, move all C’s to right side

4. find C using initial condition

5. write explicit y=f(x) or implicit f(x,y)=0 equation

6. solve for a variable if needed