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

In a BVP, conditions are imposed at the boundaries of an interval (e.g., y(0) = 0 and y(1) = 1) rather than all at the same starting point.

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 zero solution is asymptotically stable if all eigenvalues have negative real parts, unstable if any eigenvalue has positive real part.

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

By Dirichlet's theorem, the Fourier series converges to f(x) at points of continuity if f is piecewise smooth on the interval.

Question 4

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

Accepted Answer

Ae^x

Answer

Since e^x is not a solution of the homogeneous equation y'' + y = 0 (which has solutions cos x, sin x), we try y_p = Ae^x directly.

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

A saddle point has one positive and one negative real eigenvalue; trajectories along the stable manifold approach while others diverge.

BMath Bachelor of Mathematics Practice Test