Vocab
What is the order of the following ODE?
3(dy)/(dt)+4(d^3y)/(dt^3)=sin(4t)
3
The following autonomous ODE system has infinite equilibrium solutions:
x'(t)=y
y'(t)=-sin(x)
True
Solve the ODE:
dy/dt+1/2y=1/2e^{t/3}
y(t)=3/5e^{{5t}/6}+ce^{t/2}
Do the matrix multiplication:
[[0,1],[2,2]][[1],[2]]
[[2],[6]]
find the eigenvalues of the matrix
[[2,1],[1,0]]
1+-sqrt(2)
What are the steps to solving an ODE using the Laplace transform?
1) Take the Laplace transform of the equation
2) Solve for Y(s)
3) Take the inverse Laplace transform.
Sketch direction fields for
1) An unstable node
2) a saddle point
3) a stable spiral point
Which of the following ODE's are Linear:
ty''+y=sin(t)
y' = t^2y
y''+sin(y) = 0
1st and 2nd are linear, 3rd is non-linear
Does the following ODE model resonance?
2x''+18x=sin(3t)
yes. Natural frequency is sqrt(k/m)=3, forced frequency is also 3.
Find an implicit solution to the separable equation:
y'(x)=(3x^2+4x+2)/(2(y-1))
y^2 −2y = x^3 +2x^2 +2x +c
Matrix A is (4x4), Matrix B is (1x4) and vector y is (4x1). What size is the matrix
BA\vec{y}
1x1
The eigenvalues of A (2x2) are imaginary complex conjugates. What is the real-valued general solution in this case?
\vec{x}(t)=c_1e^{\lambdat}(\vec{a}cos(\mut)-\vec{b}sin(\mut))+c_2e^{\lambdat}(\vec{a}sin(\mut)+\vec{b}cos(\mut))
Using the table, what is
\mathcal{L}\{t^2e^{-t}\}
2/(s+1)^3
Draw a direction field for the ODE, including the "phase line":
x'=x(10-x)
Show on MATLAB. Should have equilibria at 0 and 10.
Give an example of a 2nd order, linear, non-homogeneous ODE with constant coefficients.
ax''+bx'+cx=f(t) !=0
the trace of a matrix A, denoted tr(A), is the sum of its diagonal elements. The characteristic equation of a 2x2 matrix can be written:
\lambda^2-tr(A)\lambda+det(A)
True
det(A-\lambdaI)=(a-\lambda)(d-\lambda)-bc
=\lambda^2-(a+d)\lambda+(ad-bc)
The equation below models a mass on a spring with damping, with no forcing term. In terms of m,b, and k, when is the system overdamped i.e. when do solutions decay exponentially?
my′′ + by′ + ky = 0
This happens when we have real eigenvalues of our characteristic equation, when
b^2-4mk > 0
find the determinant of the matrix
[[0,1,0],[1,1,3],[2,1,1]]
5
Find the critical points of the system
x'(t)=(x-2)(y+1)
y'(t)=x-xy
2 points,
(x,y) = (2,1)
(x,y) = (0,-1)
Do the integration by parts to find the Laplace transform of the function
f(t)=t
Use integration by parts. Should find
F(s)=1/s^2
The table of critical points calls all stable equilibrium "asymptotically stable" except a center which is just "stable". What is different about these terms?
Asymptotically stable means trajectories approach the point but never reach it, stable means the trajectories stay within a certain distance of the critical point but don't approach or move away from it.
Is the following equation exact? Show that it is.
2xcos(x^2+y)dx+cos(x^2+y)dy=0
Yes. An exact equation is of the form M(x,y)+N(x,y)dy/dx=0, or sometimes written as M(x,y)dx+N(x,y)dy=0. It is exact if dM/dy=DN/dx. (Show this is true)
is the y(t) a solution to the ODE?
y(t)=e^{-t}([[-1],[0]]+t[[2],[1]])
\vec{x}'(t)=[[-3,4],[-1,1]]\vec{x}
True
Find a Particular solution to the ODE:
y''+2y'+y=cos(2t)
y_p(t)=4/25sin(2t)-3/25cos(2t)
3 is an eigenvalue of the below matrix. What is the corresponding eigenvector?
[[1,2],[0,3]]
[[1],[1]]
Classify the type of critical point at (0,0)
{(x'=x-2y+xy),(y'=x+4y+y^2):}
eigenvalues of Jacobian are 3 and 2, so critical point is an unstable node
Find the inverse Laplace transform
\mathcal{L}^{-1}\{{s-1}/{s^2-s-2}\}
Use Partial fraction decomposition and find
y(t) = 1/3e^{2t}+2/3e^-t
Attempt to sketch a 3-D direction field of a critical point with 1 positive, real valued eigenvalue and 2 complex conjugate eigenvalues.
swirly in a plane, shoots of on another direction.