Differential Equations
True or False?
...
The two fundamental solutions that arise when considering an IVP
1) Does a solution exist? 2) If a solution exists, is it unique?
What are the two important assumptions when stating definitions and theorems about linear equations on some interval I?
1) The coefficient functions ai(x),i=0,1,2,...,n and g(x) are continuous 2) an(x) cannot equal zero for every x in the interval
Corollaries to the superposition principle
1) a constant multiple y=c1y1(x) of a solution y1(x) of a homogeneous linear DE is also a solution 2) A homogeneous linear DE always possesses the trivial solution y=0
Particular Solution
Any function, yp, free of arbitrary parameters that satisfies (double triangle)
Fundamental Set of Solutions
Any set y1,y2,...yn, of n linearly independent of the homogeneous linear nth order DE on an interval I on the interval
In a logistics equation, the constant K is called...
Carrying Capacity
Two questions arise when approaching a DE
Does it have a solution and is it unique?
True or False? A BVP can have exactly two solutions
False
True or False? A set of two functions f1(x) and f2(x) is linearly dependent when neither function is a constant multiple of the other on the interval
False
True or False? If a set of two functions is linearly independent, then one function is a constant multiplier of the other
False
True or False? If y1,y2,...yn form a FSS of an nth order DE on an interval I, then y1,y2,...yn must be linearly dependent
False
True or False? In the competition model dx/dt = ax-bxy, dy/dt=cy-dxy, where x(t) and y(t) are the popultions of the competing species moose and deer, respectively, the coefficient c represents the moose growth rate
False
True or False? In the growth model dP/dt=50e^0.5t, the growth constant is 50
False
True or False? The Wronskian of the function e^x and e^3x is 2e^2x
False
True or False? There does not exist a fundamental set of solutions for the homogeneous linear nth order DE on the interval I
False
True or False? A differential operator can be linear or non linear
False, linear only
True or False? Solutions of a Linear DE are not always unique
False. They are always unique
How do you find the general solution to a Homogeneous DE?
Find the Wronskian. If the Wronskian equals a number, then the given y values form a F.S.S
List the steps to these directions: y=c1x +c2xlnx is a solution of xy''-xy'=0 on (0,inf). Find a member of the family of solutions that satisfies the initial conditions y(1)=5 and y'(1)=3
Find y' Plug in the initial condition to solve for the constants c1 and c2
List the steps to these directions: Verify that the given two- parameter family of functions is the general solution of the non homogeneous differential equation on the indicated interval
Let y1 equal the stuff behind c1 and y2 equal the stuff behind c2 Derive as many times as need to see if it satisfies the given equation Take the Wronskian of y1 and y2, respectively in the columns To be a F.S.S, the Wronskian cannot equal zero
Criterion for Linearly Independent solutions
Let y1,y2,....yn be n solutions of the homogeneous linear nth order DE on an interval I. The W(y1,y2,...yn) cannot equal zero for every x in the interval
When the Wronskian = 0, then is it linear dependent or independent?
Linear Dependent
BVP
Linear differential equation of 2nd order or greater in which the dependent variable y or its derivatives are specified at different points
List the steps to these directions: Verify that the given functions form a F.S.S of the differential equation on the indicated interval. Form the general solution.
Set y equal to each value given Find the number of derivatives to match the order of the DE Plug the derivatives into the given equation and verify that they each equal zero, which makes them a F.S.S Use the Wronskian to determine if they equal zero
In the logistic model for population growth dP/dt=P(8-2P), the carrying capacity of the population is
k=a/b so in this case, 4
Temperature Formula
T=Ce^kt+Tm, where Tm is your temperature medium
Superposition Principle- Homogeneous Equations states that...
The sum, or superposition, of two or more solutions of a homogeneous linear DE is also a solution
What does a differential operator do?
Transforms a differential function into another function, get it to factor
True or False? 3x^4y'' + e^xy' = (tanx)y is a homogeneous differential equation
True
True or False? A BVP may have several solutions, a unique solution, or no solution at all
True
True or False? A logistic function is the solution to a logistic equation
True
True or False? An initial Value Problem is guaranteed a solution that is unique and exists
True
True or False? Any linear differential equation can be expressed in terms of the D notation
True
True or False? Every homogeneous linear differential equation always has a solution
True
True or False? Growth and Decay models and Temperature models are linear models
True
True or False? If yp1 and yp2 are solutions of the DE y'' + P(x)y' + Q(x) =0, then y=yp1=yp2 is also a solution to the DE.
True
True or False? L is the symbol for a linear operator
True
True or False? Polynomial expressions are also differential operators
True
True or False? Solutions to initial value problem are unique
True
True or False? The differential equation dP/dt=kP is both separable and linear
True
True or False? The differential operator D is a linear operator
True
List the steps to these directions: Determine whether the given set of functions is linearly independent or linearly dependent
Use the Wronskian to determine whether the functions or independent or dependent
Growth and Decay Formula
X=Ce^kt, where X is your total or final population at any given time,t, and C is your initial population and k is your constant of proportionality
General solution
a family of solutions defined on some interval I that contains all solutions of the DE that are defined on I
What is the difference between an IVP and a BVP?
an initial value problem has all of the conditions specified at the same value of the independent variable in the equation. ex, y(0)= #, y'(0)= # whereas a BVP may have y(0)= # y'(2)= #
When discussing the logistics equation, we talked about the density dependent hypothesis. This DE is given by dP/dt=Pf(P). What is the simpliest assumption we can make about f(P)?
f(P) is linear
If a set of two functions is linear dependent, then...
then one function is simply a constant multiple of the other