Differential Equations

Question 1

What is the order of an ODE?

Answer 1

The order of its highest derivative

Question 2

When is an ODE autonomous?

Answer 2

When the independent variable doesn’t appear explicitly in the equation

Question 3

When is an ODE homogeneous?

Answer 3

When the function of the independent variable is equal to zero

Question 4

When is a function F an anti-derivative of f

Answer 4

When F’(t) = f(t)

Question 5

State the Existence and Uniqueness Theorem

Answer 5

If f(x,t) and df/dx(x,t) are continuous for a < x < b and c < t < d for any x(0) and t(0) there is a unique solution to the initial value problem

Question 6

What is the enlarged phase space

Answer 6

The space of x vs t where every point in the plane has a vector with gradient f(x,t) and length = lf(x,t)l

Question 7

When is a point of an ODE a fixed point

Answer 7

When dx/dt = 0

Question 8

When is a ODE fixed point x* stable

Answer 8

When f’(x*) < 0

Question 9

When is an ODE fixed point x* unstable

Answer 9

When f’(x*) > 0

Question 10

What is the stability of an ODE fixed point when f’(x*) = 0

Answer 10

Structurally unstable

Question 11

When are two functions x1(t) and x2(t) linearly independent

Answer 11

When the only solution to a1x1 + a2x2 = 0 is a1=a2=0

Question 12

What is the solution to a homogenous 2nd order equation with

i) two real roots (y, z)
ii) repeated real roots (y)
iii) complex roots (p + iq)

Answer 12

i) Aexp(yt) + Bexp(zt)
ii) (A+Bt)exp(kt)
iii) exp(pt)(Acosqt + Bsinqt)

Question 13

What is Newtons II law

Answer 13

Force = mass x acceleration

Question 14

What is the equation of a mass/spring system with friction?

Answer 14

m(d^2x/dt^2) + c(dx/dt) + kx = 0

Question 15

Under what circumstances do we achieve SHM?

Answer 15

c = 0

m(d^2x/dt^2) + kx = 0

Question 16

For the equation
m(d^2x/dt^2) + c(dx/dt) + kx = 0
When is it undamped?

Answer 16

c = 0

Question 17

For the equation
m(d^2x/dt^2) + c(dx/dt) + kx = 0
When is it under-damped?

Answer 17

c^2 - 4mk < 0

Question 18

For the equation
m(d^2x/dt^2) + c(dx/dt) + kx = 0
When is it over-damped?

Answer 18

c^2 - 4mk > 0

Question 19

How does m(d^2x/dt^2) + c(dx/dt) + kx = 0 change when there is forcing

Answer 19

The right hand side is equal to Fcos((omega)t) and k is set to equal w^2

Question 20

With no forcing and no friction, what is the natural frequency of the mass spring system with forcing

Answer 20

w/2pi

Question 21

What is the order of a difference equation?

Answer 21

The difference between the highest and lowest index of x

Question 22

What is the solution to the difference equation x(n+1) =ax(n)

Answer 22

x(n) = a^n x(0)

Question 23

What is the solution to a second order difference equation with

i) Two real roots (y,z)
ii) Repeated real roots (k)
iii) Complex roots (p+iq)

Answer 23

i) x(n) = A(y)^n + B(z)^n
ii) x(n) = A(k)^n + Bn(k)^n
iii) x(n) = r^n(Acosn(theta) + Bsinn(theta))
where r = l p+iq l theta = arctan(p/q)

Question 24

When is a point x* a fixed point of a difference equation

Answer 24

When f(x)=x

Question 25

When is a point x* a stable fixed point of a difference equation

Answer 25

when l f’(x*) l < 1

Question 26

When is a point x* an unstable fixed point of a difference equation

Answer 26

when l f’(x*) l > 1

Question 27

When does an equation have a period two orbit

Answer 27

When it tends to having two alternating points

i.e. f(f(x))=x

Question 28

When does a system of first-order ODE’s have a unique solution

Answer 28

When all first order partial derivatives are continuous functions

Question 29

What is the general solution for distinct real roots to dx/dt = Ax where x is a vector

Answer 29

x(t) = Av1exp(k1t) + Bv2exp(k2t)

where v1 and v2 are the eigenvectors of the eigenvalues k1 and k2

Question 30

What is the general solution for complex roots to dx/dt = Ax where x is a vector

Answer 30

x(t) = exp(pt)((acosqt+bsinqt)v1 + (bcosqt-asinqt)v2)
Where Eigenvector = v1 + iv2
Eigenvalue k=p+iq

Question 31

What is the stability of the origin if eigenvalues k1, k2 are both negative

Answer 31

Stable

Question 32

What is the stability of the origin if eigenvalues k1, k2 are both positive

Answer 32

Unstable

Question 33

What is the stability of the origin if eigenvalues k1 < 0, k2 > 0

Answer 33

Saddle point

Question 34

How do you uncouple a system of equations with distinct eigenvalues

Answer 34

Take P with columns v1,v2. P-1AP = matrix B with entries eigenvalues along the main diagonal. So dy/dt = By

Question 35

When are complex solutions to a system of equations stable

Answer 35

When the Real part of the eigenvalue is negative

Question 36

What is the shape of the phase diagram of a solution with complex eigenvalues

Answer 36

A spiral into the origin with direction

Question 37

What is the solution to a system of equations with a repeated eigenvalue k and eigenvalue v

Answer 37

x(t) = Bvexp(kt) + C(a+tv)exp(kt)

a is a vector sick that (A-kI)a = v

Question 38

At a point (x,y) the vector (df/dx, df/dy) at (x,y) is normal to the level curve at (x,y). Prove

Answer 38

On a level set df/dt = 0. Use chain rule on df/dt to get df/dt = df/dx dx/dt + df/dy dy/dt = 0.
This is grad f times (dx/dt dy/dt). If thats zero then its perpendicular to the tangent and thus is a normal

Question 39

The maximum value of the directional vector f(x,y) occurs in the direction of grad f with maximum value l grad f(x,y) l. Prove

Answer 39

Let theta be the angle between grad f(x,y) and v.
Directional vector f(x,y) = grad f(x,y) unit v
= grad f(x,y) unit v cos(theta)
= grad f(x,y) cos(theta)
has a max at theta = 0

Question 40

What is the general formula for mixing problems, where x is the amount of salt in a tank

Answer 40

dx/dt = (concentration in of salt)(rate of water in) - (x/total volume)(rate of water out)