Differential Equations
A 1st order equation (meaning only x', not x') always has the form... For example, x'=
tˆ2(x) has the form f(t,x) - ------------ANSWER: x = f(t,x), f(t,x) = tˆ2(x)
cos(0), cos(pi), cos(2pi), cos(pi/2). Intersects origin? - ------------ANSWER: 1, -1, 1,
0. No.
Do solutions cross? - ------------ANSWER: No!
dy/dx = y. Solution(s)? - ------------ANSWER: y(x) = Aeˆx (because y(x) = eˆx doesn't
satisfy y'= y). A is a constant.
How do we model a logistic equation with a parameter (such as rabbit pop growth,
while the farmer kills a rabbit a day?) - ------------ANSWER: x' = x(1-x) - h
How do we solve equations of the form x'= polynomial(x)? LIke x'= x(1-x) -
------------ANSWER: 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
How to solve a calc 1 style equation like x = tˆ2 + 2t + 1?? - ------------ANSWER:
Take anti derivative x = f(tˆ2 + 2t + 1dt), x(t) = tˆ3 /3 + tˆ2 + t + C
How to solve an autonomous equation? x'= x - ------------ANSWER: 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
How to solve axˆ2 + bx + c = 0?? - ------------ANSWER: x equals -b plus or minus
square root of (bˆ2 - 4ac) all over 2a
In an autonomous equation, what does the slope look like? Why? -
------------ANSWER: The slope field is constant along horizontal lines, because it
doesn't depend on t.
Log (u)ˆn = - ------------ANSWER: nlog(u)
Log(u/v) = - ------------ANSWER: Log(u) - Log(v)
Log(uv) = - ------------ANSWER: Log(u) + Log(v)
sin(0), sin(pi), sin(2pi), sin(pi/2). Intersects origin? - ------------ANSWER: 0, 0, 0, 1.
Yes.
, The 2 steps to sketching the bifucation diagram for a formua like x'=xˆ2 - ax -
------------ANSWER: 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 are the 3 types of diff equations? - ------------ANSWER: 1) Calc I style x'=(t
only) 2) Autonomous x=(x only) 3) Separate equations x'= (x stuff only) + (t stuff only)
What do logistic equations model? - ------------ANSWER: They model population
growth with a pop. cap.
What do we mean by bifurcation? What creates it? - ------------ANSWER: 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.
What does f(t,x) tell us? - ------------ANSWER: The slope of the line at a point t,x
What is a differential equation? - ------------ANSWER: An equation which relates
several quantities and their (1st, 2nd, 3rd, etc) derivatives
What is a phase line? - ------------ANSWER: A simplified slope field for autonomous
equations.
What is a set of phase lines together called? - ------------ANSWER: A phase portrait
or a bifurcation diagram
What is Euler's method? - ------------ANSWER: Euler method, named after Leonhard
Euler, is a first-order numerical procedure for solving ordinary differential equations
(ODEs) with a given initial value.
What is the basic logistic equation model? Exponential and with cap? Simplified? -
------------ANSWER: x' = ax
x' = a(x(1 - x/n)
x'= x(1-x)
x = bˆy is equivalent to - ------------ANSWER: y = logb(X)
x'= √x. Solution? - ------------ANSWER: 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? - ------------ANSWER: x = tan(t + C) -- dx/dt = 1 + xˆ2 -- (dx/1 +
xˆ2) = dt -- ∫(dt/1 + xˆ2) = ∫dt -- arctan(x) = t + c -- x = tan(t + C)
A 1st order equation (meaning only x', not x') always has the form... For example, x'=
tˆ2(x) has the form f(t,x) - ------------ANSWER: x = f(t,x), f(t,x) = tˆ2(x)
cos(0), cos(pi), cos(2pi), cos(pi/2). Intersects origin? - ------------ANSWER: 1, -1, 1,
0. No.
Do solutions cross? - ------------ANSWER: No!
dy/dx = y. Solution(s)? - ------------ANSWER: y(x) = Aeˆx (because y(x) = eˆx doesn't
satisfy y'= y). A is a constant.
How do we model a logistic equation with a parameter (such as rabbit pop growth,
while the farmer kills a rabbit a day?) - ------------ANSWER: x' = x(1-x) - h
How do we solve equations of the form x'= polynomial(x)? LIke x'= x(1-x) -
------------ANSWER: 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
How to solve a calc 1 style equation like x = tˆ2 + 2t + 1?? - ------------ANSWER:
Take anti derivative x = f(tˆ2 + 2t + 1dt), x(t) = tˆ3 /3 + tˆ2 + t + C
How to solve an autonomous equation? x'= x - ------------ANSWER: 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
How to solve axˆ2 + bx + c = 0?? - ------------ANSWER: x equals -b plus or minus
square root of (bˆ2 - 4ac) all over 2a
In an autonomous equation, what does the slope look like? Why? -
------------ANSWER: The slope field is constant along horizontal lines, because it
doesn't depend on t.
Log (u)ˆn = - ------------ANSWER: nlog(u)
Log(u/v) = - ------------ANSWER: Log(u) - Log(v)
Log(uv) = - ------------ANSWER: Log(u) + Log(v)
sin(0), sin(pi), sin(2pi), sin(pi/2). Intersects origin? - ------------ANSWER: 0, 0, 0, 1.
Yes.
, The 2 steps to sketching the bifucation diagram for a formua like x'=xˆ2 - ax -
------------ANSWER: 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 are the 3 types of diff equations? - ------------ANSWER: 1) Calc I style x'=(t
only) 2) Autonomous x=(x only) 3) Separate equations x'= (x stuff only) + (t stuff only)
What do logistic equations model? - ------------ANSWER: They model population
growth with a pop. cap.
What do we mean by bifurcation? What creates it? - ------------ANSWER: 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.
What does f(t,x) tell us? - ------------ANSWER: The slope of the line at a point t,x
What is a differential equation? - ------------ANSWER: An equation which relates
several quantities and their (1st, 2nd, 3rd, etc) derivatives
What is a phase line? - ------------ANSWER: A simplified slope field for autonomous
equations.
What is a set of phase lines together called? - ------------ANSWER: A phase portrait
or a bifurcation diagram
What is Euler's method? - ------------ANSWER: Euler method, named after Leonhard
Euler, is a first-order numerical procedure for solving ordinary differential equations
(ODEs) with a given initial value.
What is the basic logistic equation model? Exponential and with cap? Simplified? -
------------ANSWER: x' = ax
x' = a(x(1 - x/n)
x'= x(1-x)
x = bˆy is equivalent to - ------------ANSWER: y = logb(X)
x'= √x. Solution? - ------------ANSWER: 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? - ------------ANSWER: x = tan(t + C) -- dx/dt = 1 + xˆ2 -- (dx/1 +
xˆ2) = dt -- ∫(dt/1 + xˆ2) = ∫dt -- arctan(x) = t + c -- x = tan(t + C)
