Logarithms Questions and Answers

Answer: B

A first-order equation is separable when it can be rearranged so that each variable sits with its own differential:

\[ \frac{dy}{dx} = f(x)g(y) \quad\Longrightarrow\quad \frac{dy}{g(y)} = f(x)\,dx \]

Each side can then be integrated on its own. Separability is precisely the property that the variables can be pulled apart in this way.

Answer: C

Prime notation is convenient only for low orders:

\[ y', \quad y'', \quad y''' \]

Beyond the third derivative the primes become hard to count, and the numeral form \( y^{(4)} \) or the Leibniz form \( \tfrac{d^{4}y}{dx^{4}} \) is used instead.

Answer: D

The value of a variable at the designated starting time \( t = 0 \) is the seed value - also called the initial value.

It is what an initial condition supplies, and it is what selects one particular solution from the general family.

Answer: A

Some elementary functions have antiderivatives that cannot be written as elementary functions at all.

\[ \int e^{-x^{2}}\,dx, \qquad \int \frac{\sin x}{x}\,dx \]

Both integrals exist, but neither can be expressed with finitely many algebraic, exponential, logarithmic and trigonometric operations. New named functions - the error function, the sine integral - are defined for them.

Answer: A

A deterministic model is one in which the present state is fixed entirely by the parameters and the earlier states: run it twice from the same start and it gives the same answer twice.

A stochastic or probabilistic model includes randomness, so repeated runs can differ.

Answer: C

A floating model rests on neither theory nor observation - it simply assumes a structure that seems plausible.

A deductive model is derived from theory and an inductive one built up from data; the floating model has neither footing, which is why its conclusions carry little weight on their own.

Answer: A

In the Laplace-transform method, the very first step is to apply the transform to the whole differential equation, turning derivatives in \( t \) into powers of \( s \):

\[ \mathcal{L}\{y'\} = sY - y(0), \qquad \mathcal{L}\{y''\} = s^{2}Y - sy(0) - y'(0) \]

The result is an algebraic (subsidiary) equation in \( Y(s) \). Solving it is step 2 and inverting is step 3.

Answer: B

Writing derivatives with dashes - \( y' \), \( y'' \), \( y''' \) - is prime notation, due to Lagrange.

Leibniz notation is the alternative, \( \tfrac{dy}{dx} \), \( \tfrac{d^{2}y}{dx^{2}} \), which has the advantage of naming the independent variable explicitly.

Answer: B

Apply the chain rule to \( g(x) = \sin(2x^{2}+x) \).

\[ \frac{d}{dx}\sin u = \cos u \cdot \frac{du}{dx} \]

With \( u = 2x^{2}+x \):

\[ \frac{du}{dx} = 4x + 1 \]

\[ g'(x) = (4x+1)\cos(2x^{2}+x) \]

The outer function changes from sine to cosine; leaving it as a sine is the error the distractors are built on.

Answer: D

The standard transform pair is

\[ \mathcal{L}\{\sin \omega t\} = \frac{\omega}{s^{2}+\omega^{2}} \]

while its companion is

\[ \mathcal{L}\{\cos \omega t\} = \frac{s}{s^{2}+\omega^{2}} \]

The two differ only in the numerator: \( \omega \) for the sine, \( s \) for the cosine. That is the pair most often confused.