BMath Bachelor of Mathematics Differential Equations 2

Question 1

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

Accepted Answer

1/(s − a)

Answer

1/(s + a)

Answer

a/(s² + a²)

Answer

s/(s² + a²)

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

e^(∫Q(x) dx)

Answer

1/P(x)

Answer

Q(x)/P(x)

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

Asymptotically stable — solutions spiral inward

Answer

Unstable — solutions spiral outward

Answer

A saddle point

Question 4

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

Accepted Answer

y = C₁ cos(2x) + C₂ sin(2x)

Answer

y = C₁e^(2x) + C₂e^(−2x)

Answer

y = (C₁ + C₂x)e^(2x)

Answer

y = C₁ cosh(2x) + C₂ sinh(2x)

Question 5

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

Accepted Answer

dy/dx = f(x)·g(y)

Answer

dy/dx = f(x) + g(y)

Answer

y'' + y = 0

Answer

y' = ay + b

Question 6

The convolution theorem for Laplace transforms states that ℒ{f * g} equals:

Accepted Answer

ℒ{f} · ℒ{g}

Answer

ℒ{f} + ℒ{g}

Answer

ℒ{f} / ℒ{g}

Answer

ℒ{f} − ℒ{g}