Chapter 1 Flashcards
Derivative
Definition (words)
the derivative of a function y(x) at a particular value of x is the slope of the tangent to the curve at the point (x,y(x))
Derivative
Definition (equation)
dy/dx = (𝛿x->0)lim y(x+𝛿x)-y(x) / 𝛿x
Ordinary Differential Equation
Definition
an equation linking a dependent variable y(x), an independent variable x, and certain derivatives
Implicit Solution
Definition
a solution that contains undefined constants
Explicit Solution
Definition
a solution that doesn’t have any undefined constants
Order
Definition
the order of an ODE is the largest number of times that the dependent variable is differentiated in the ODE
Linear
Definition
an ODE is linear if it contains no products or powers greater than one of the dependent variable or its derivatives
in a linear ODE the independent variable can be any power
Autonomous
Definition
an ODE is autonomous if there is no explicit mention of the independent variable
General Solution
Definition
an implicit solution to a differential equation, an nth order ODE will contain n arbitrary constants
Solution to Equation
Method 1
-assume equation of the form
y’’ + Ay’ + By = 0
-differentiate solution to find y’ and y’’
-substitute into the assumed form
-collect terms in the constants from the solution together (C1, C2, …)
-solve for A and B
Solution to Equation
Method 2
- rearrange the solution so you have constants on one side and variables on the other
- differentiate
- constants will become 0
- if you have more constants left, repeat until they are all 0
Initial Value Problem
Definition
for an ODE of order n, we are given n values of y or its derivatives at a single value of x
Boundary Value Problem
Definition
for an ODE of order n, we are given n values of y or its derivatives at, at least, two values of x
