Ordinary Differential Equations and Linear Algebra

What principle says that if y1 and y2 are solutions to a linear homogeneous equation, so is c1y1+c2y2?

Superposition principle

How would you rewrite Ax=b with the inverse of A?

x=A^(-1)b

Picard’s theorem says that if _(a)__ is continuous in the neighborhood of a point, a solution exists. If _(a)__ is continuous in the neighborhood of a point and _(b)__ is continuous in the neighborhood of a point, the solution is unique. What are a and b?

f, partial derivative of f with respect to y

Matrix is not invertible when det = ?

0

What is the equation for the integrating factor?

u = e^integral(p(t)dt)

What is the new equation once integrating factor is implemented?

(u*y)’=u*f(x)

How do you calculate the Jacobian of two functions, x’ and y’, both of which have variables x and y? J=

[partial derivative of x’ with respect to x, partial derivative of x’ with respect to y; partial derivative of y’ with respect to x, partial derivative of y’ with respect to y]

If both eigenvalues are real and positive, it is:

unstable source

If both eigenvalues are real and negative:

stable sink

If one eigenvalue is positive and one is negative:

unstable saddle point

If eigenvalues are positive and repeated:

unstable source

If eigenvalues are negative and repeated:

stable sink

If eigenvalues are 0 and repeated:

inconclusive

If eigenvalues are complex and the real component is negative:

stable spiral sink

If eigenvalues are complex and real component is positive:

unstable spiral source

If eigenvalues are complex and real component is 0:

Neutrally stable center

What is a nullcline?

Line where at least one derivative is 0

What is the rank of a matrix?

number of linearly independent columns

What is the nulity of a matrix?

number of columns - rank

What is the column space of a matrix?

span of all columns

What is the null space of a matrix?

Span of all linearly dependent columns

Formula for homogeneous solution of first order linear differential? yh = ?

Ce^-integral(p(t)dt)