Differential Equations Flashcards
What is the order of an ODE?
The order of its highest derivative
When is an ODE autonomous?
When the independent variable doesn’t appear explicitly in the equation
When is an ODE homogeneous?
When the function of the independent variable is equal to zero
When is a function F an anti-derivative of f
When F’(t) = f(t)
State the Existence and Uniqueness Theorem
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
What is the enlarged phase space
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
When is a point of an ODE a fixed point
When dx/dt = 0
When is a ODE fixed point x* stable
When f’(x*) < 0
When is an ODE fixed point x* unstable
When f’(x*) > 0
What is the stability of an ODE fixed point when f’(x*) = 0
Structurally unstable
When are two functions x1(t) and x2(t) linearly independent
When the only solution to a1x1 + a2x2 = 0 is a1=a2=0
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)
i) Aexp(yt) + Bexp(zt)
ii) (A+Bt)exp(kt)
iii) exp(pt)(Acosqt + Bsinqt)
What is Newtons II law
Force = mass x acceleration
What is the equation of a mass/spring system with friction?
m(d^2x/dt^2) + c(dx/dt) + kx = 0
Under what circumstances do we achieve SHM?
c = 0
m(d^2x/dt^2) + kx = 0
For the equation
m(d^2x/dt^2) + c(dx/dt) + kx = 0
When is it undamped?
c = 0
For the equation
m(d^2x/dt^2) + c(dx/dt) + kx = 0
When is it under-damped?
c^2 - 4mk < 0
For the equation
m(d^2x/dt^2) + c(dx/dt) + kx = 0
When is it over-damped?
c^2 - 4mk > 0
How does m(d^2x/dt^2) + c(dx/dt) + kx = 0 change when there is forcing
The right hand side is equal to Fcos((omega)t) and k is set to equal w^2
With no forcing and no friction, what is the natural frequency of the mass spring system with forcing
w/2pi
What is the order of a difference equation?
The difference between the highest and lowest index of x
What is the solution to the difference equation x(n+1) =ax(n)
x(n) = a^n x(0)
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)
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)
When is a point x* a fixed point of a difference equation
When f(x)=x
When is a point x* a stable fixed point of a difference equation
when l f’(x*) l < 1
When is a point x* an unstable fixed point of a difference equation
when l f’(x*) l > 1
When does an equation have a period two orbit
When it tends to having two alternating points
i.e. f(f(x))=x
When does a system of first-order ODE’s have a unique solution
When all first order partial derivatives are continuous functions
What is the general solution for distinct real roots to dx/dt = Ax where x is a vector
x(t) = Av1exp(k1t) + Bv2exp(k2t)
where v1 and v2 are the eigenvectors of the eigenvalues k1 and k2
What is the general solution for complex roots to dx/dt = Ax where x is a vector
x(t) = exp(pt)((acosqt+bsinqt)v1 + (bcosqt-asinqt)v2)
Where Eigenvector = v1 + iv2
Eigenvalue k=p+iq
What is the stability of the origin if eigenvalues k1, k2 are both negative
Stable
What is the stability of the origin if eigenvalues k1, k2 are both positive
Unstable
What is the stability of the origin if eigenvalues k1 < 0, k2 > 0
Saddle point
How do you uncouple a system of equations with distinct eigenvalues
Take P with columns v1,v2. P-1AP = matrix B with entries eigenvalues along the main diagonal. So dy/dt = By
When are complex solutions to a system of equations stable
When the Real part of the eigenvalue is negative
What is the shape of the phase diagram of a solution with complex eigenvalues
A spiral into the origin with direction
What is the solution to a system of equations with a repeated eigenvalue k and eigenvalue v
x(t) = Bvexp(kt) + C(a+tv)exp(kt)
a is a vector sick that (A-kI)a = v
At a point (x,y) the vector (df/dx, df/dy) at (x,y) is normal to the level curve at (x,y). Prove
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
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
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
What is the general formula for mixing problems, where x is the amount of salt in a tank
dx/dt = (concentration in of salt)(rate of water in) - (x/total volume)(rate of water out)
