BMath Bachelor of Mathematics Differential Equations 2

Question 1

The Laplace transform ℒ{e^(at)} equals:

Accepted Answer

1/(s − a)

Answer

By definition, ℒ{e^(at)} = ∫₀^∞ e^(at) e^(−st) dt = 1/(s − a) for s > a.

Question 2

A first-order linear ODE dy/dx + P(x)y = Q(x) is solved using an integrating factor μ(x) equal to:

Accepted Answer

e^(∫P(x) dx)

Answer

Multiplying by μ(x) = e^(∫P(x) dx) makes the left side an exact derivative d/dx[μy], enabling direct integration.

Question 3

What type of equilibrium point is a center in a system of ODEs?

Accepted Answer

Stable but not asymptotically stable — solutions form closed orbits

Answer

A center has purely imaginary eigenvalues; nearby trajectories form closed orbits and the equilibrium is stable but not attracting.

Question 4

The general solution of y'' + 4y = 0 is:

Accepted Answer

y = C₁ cos(2x) + C₂ sin(2x)

Answer

The characteristic roots are r = ±2i (purely imaginary), giving the general solution y = C₁cos(2x) + C₂sin(2x).

Question 5

Separation of variables is applicable to an ODE of the form:

Accepted Answer

dy/dx = f(x)·g(y)

Answer

When the ODE has the product form dy/dx = f(x)g(y), we can separate to get dy/g(y) = f(x) dx and integrate each side.

BMath Bachelor of Mathematics Practice Test