BMath Bachelor of Mathematics Differential Equations 3

Question 1

A boundary value problem differs from an initial value problem in that:

Accepted Answer

Conditions are specified at two different points rather than all at one point

Answer

It involves higher-order derivatives

Answer

It has no solutions

Answer

It requires numerical methods

Question 2

For the system x' = Ax with constant matrix A, the stability of the equilibrium x = 0 depends on:

Accepted Answer

The real parts of the eigenvalues of A

Answer

The trace of A

Answer

The determinant of A

Answer

Whether A is symmetric

Question 3

The Fourier series of a function f on [−π, π] converges to f(x) at every point where:

Accepted Answer

f is piecewise smooth and f is continuous at x

Answer

f is differentiable

Answer

f is continuous on all of ℝ

Answer

f is bounded

Question 4

What is the particular solution form to try for y'' + y = e^x using undetermined coefficients?

Accepted Answer

Ae^x

Answer

Axe^x

Answer

A cos x + B sin x

Answer

A + Be^x

Question 5

The phase portrait of a saddle point in a 2D linear system shows:

Accepted Answer

Some trajectories approaching and others moving away from the equilibrium

Answer

Closed orbits around the equilibrium

Answer

All trajectories spiraling toward the origin

Answer

All trajectories moving parallel to each other

Question 6

Which of the following is an exact differential equation?

Accepted Answer

(2x + y) dx + x dy = 0

Answer

(2xy) dx + (x² + y²) dy = 0

Answer

(x + y) dx − x dy = 0

Answer

(y²) dx + (x²) dy = 0