Quiz 1 Review Flashcards
(20 cards)
Differential equation
An equation relating an unknown function and one or more of its derivatives.
Order
the order of a differential equation is the order of the highest derivative that appears in it.
Ordinary differential equation
The unknown function (dependent variable) depends on only a single independent variable.
General solution
A description of all possible solutions to a DE (usually has a parameter).
Particular Solution
A solution to an IVP
Existence and uniqueness theorem - Preface
Consider dy/dx=f(x,y) and a point (a,b).
Existence and uniqueness theorem Part i
i) if f(x,y) is continuous in an open rectangle containing (a,b), then the IVP dy/dx=f(x,y), y(a)=b has at least one solution. The domain of the solution will include an open interval containing a.
Existence and uniqueness theorem part ii
ii) if df/dy is continuous in an open rectangle containing (a,b), then the IVP dy/dx=f(x,y), y(a)=b has a unique solution.
Seperable
Separable provided that H(x,y) can be written as the product of a function of x and a function of y.
Linear first-order equation
dy/dx + P(x)y=Q(x)
What is the integration factor?
e∫P(x)dx
Integration factor equation when multiplying by IF
IF*y=∫IF*Qdx
First order linear DE’s can be . . . (L,S, four things)
Linear and separable
Separable, not linear
linear not separable
neither
Autonomous DE’s
DE’s of the following form: dy/dx=f(y), rate of change only depends on the current state of the system.
Phase Diagram
Indicates the direction of change in x as a function of x itself.
Equilibrium Solution
A constant solution of a differential equation.
Where are critical points?
when f(x)=0
Euler’s Method
yn+1=yn+f(xn,yn)h
Improved Euler Method
un+1=yn+f(xn,yn)h
yn+1=yn + 1/2(f(xn,yn) + f(xn+1,un+1)h
