Differential Equations

What is a differential equation?

An equation which relates several quantities and their (1st, 2nd, 3rd, etc) derivatives

y' = 0. What is the rate of change of y? What are the solution(s)?

Nothing. Y is a constant quantity. The solutions are y(x) = C, where C is a constant.

y' = x. Solution(s)?

dy/dx = x, so y(x) = 1/2(xˆ2) + C

y = 2x + [cos(xˆ2)]. Solution(s)?

sinxˆ2 + C

dy/dx = y. Solution(s)?

y(x) = Aeˆx (because y(x) = eˆx doesn't satisfy y'= y). A is a constant.

A 1st order equation (meaning only x', not x') always has the form... For example, x'= tˆ2(x) has the form f(t,x)

x = f(t,x), f(t,x) = tˆ2(x)

What does f(t,x) tell us?

The slope of the line at a point t,x

What are the 3 types of diff equations?

1) Calc I style x'=(t only) 2) Autonomous x=(x only) 3) Separate equations x'= (x stuff only) + (t stuff only)

How to solve a calc 1 style equation like x = tˆ2 + 2t + 1??

Take anti derivative x = f(tˆ2 + 2t + 1dt), x(t) = tˆ3 /3 + tˆ2 + t + C

How to solve an autonomous equation? x'= x

dx/dt = x. Shuffle x to one side, dt to the other. dx/x = dt. Take the integral. ∫dx/x = ∫dt -->log(x) = t+C. Solve for x. eˆ(logx) = eˆ(t +C). X = eˆ(t+C) = eˆt * eˆc = Aeˆt

x'= √x. Solution?

x = (tˆ2)/4 + Ct + Cˆ2. HOW? dx/dt = sqrt(x) -- dx/(sqrt(x)) = ∫dt -- ∫xˆ(-1/2)dx = t + C -- 2xˆ(1/2) = t+ C -- xˆ(1/2) = (t/2) + (C/2) -- x = (t/2 + C)ˆ2 = tˆ2 /4 + Ct + Cˆ2

x'= 1 + xˆ2? Solution?

x = tan(t + C) -- dx/dt = 1 + xˆ2 -- (dx/1 + xˆ2) = dt -- ∫(dt/1 + xˆ2) = ∫dt -- arctan(x) = t + c -- x = tan(t + C)

x'= t * x. Solution?

dx/dt = tx -- ∫dx/x = ∫tdt -- logx = (tˆ2)/2 + C -- x = eˆ(tˆ2 /2 + C) = Aeˆ((tˆ2)/2)

What is a phase line?

A simplified slope field for autonomous equations.

In an autonomous equation, what does the slope look like? Why?

The slope field is constant along horizontal lines, because it doesn't depend on t.

Do solutions cross?

No!

What do logistic equations model?

They model population growth with a pop. cap.

What is the basic logistic equation model? Exponential and with cap? Simplified?

x' = ax

x' = a(x(1 - x/n)

x'= x(1-x)

How do we model a logistic equation with a parameter (such as rabbit pop growth, while the farmer kills a rabbit a day?)

x' = x(1-x) - h

What is a set of phase lines together called?

A phase portrait or a bifurcation diagram

How to solve axˆ2 + bx + c = 0??

x equals -b plus or minus square root of (bˆ2 - 4ac) all over 2a

What do we mean by bifurcation? What creates it?

The qualitative change in the solution when the parameter changes in certain differential equations. It is created by a new equilibrium appearing or an old one disappearing.

How do we solve equations of the form x'= polynomial(x)? LIke x'= x(1-x)

dx/dt = x(1-x) -- ∫dx/x(1-x) = ∫dt -- ∫dx/x-xˆ2 = t + c -- [to find ∫dx/p(x), use partial fractions] -- ∫dx/x(1-x) = A/x + B/(1-x) = A(1-x) + B

Log(uv) =

Log(u) + Log(v)

Log(u/v) =

Log(u) - Log(v)

Log (u)ˆn =

nlog(u)

y = logb(X) is equivalent to

x= bˆy

x = bˆy is equivalent to

y = logb(X)

The 2 steps to sketching the bifucation diagram for a formua like x'=xˆ2 - ax

1) Find where xˆ2 - ax= 0 [in this case, a=x and x=0] 2) Draw those cases on the bifurcation diagram 3) Fill in the spaces in between

What is Euler's method?

Euler method, named after Leonhard Euler, is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

cos(0), cos(pi), cos(2pi), cos(pi/2). Intersects origin?

1, -1, 1, 0. No.

sin(0), sin(pi), sin(2pi), sin(pi/2). Intersects origin?

0, 0, 0, 1. Yes.