Logarithms Questions and Answers

Answer: B

\( \sin y = 1 \) is non-linear, because the unknown sits inside a sine rather than appearing to the first power on its own.

A linear equation admits the unknown only as \( ay + b \); anything else - a power, a product of unknowns, or a transcendental function of the unknown - makes it non-linear:

\[ 3y + 2 = 0 \;\;\text{linear}, \qquad \sin y = 1,\; y^{2} = 4,\; e^{y} = 3 \;\;\text{non-linear} \]

The practical consequence is that superposition fails: two solutions of a non-linear equation cannot be added to give a third.

Answer: B

\( F(x) = -c/b \) is a non homogeneous solution - a particular solution belonging to an equation whose right-hand side is not zero.

For \( by + c = 0 \) the constant \( y = -c/b \) satisfies the equation, and it exists only because of the constant term \( c \). Remove that term and the equation is homogeneous, whose solution would be \( y = 0 \).

The general solution of a non homogeneous linear equation is the sum of the two:

\[ y \;=\; y_{\text{homogeneous}} + y_{\text{particular}} \]

Answer: A

"Proportional to the number present" translates directly into a constant of proportionality times \( A \):

\[ A' = \frac{dA}{dt} = cA \]

This is the exponential growth law, whose solution is \( A(t) = A_{0}e^{ct} \). The same equation governs radioactive decay, with \( c \) negative.

Answer: C

A real number is precisely a quantity that can be placed on the number line - every point on the line corresponds to one real number and vice versa.

Natural and whole numbers occupy only isolated points on that line, leaving gaps for the fractions and irrationals between them.

Answer: B

Solving a differential equation by Laplace transform runs in three steps:

1. Transform the equation into an algebraic subsidiary equation in \( s \)
2. Solve that subsidiary equation by algebra alone
3. Take the inverse transform to recover the solution in \( t \)

Step 2 is where the method pays off: no calculus is needed, only ordinary algebraic manipulation.

Answer: D

Square brackets include the endpoint; round brackets exclude it.

\[ [a,b] = \{\, x \mid a \leq x \leq b \,\} \]

So \( [a,b] \) is the closed interval. \( (a,b) \) is open, and the two mixed forms are half-open.

Answer: B

A rate of change is a derivative, and an equation containing derivatives is a differential equation.

\[ \frac{dy}{dt} = f(t, y) \]

This is why differential equations model so much of physics, chemistry and biology - those subjects are largely statements about how things change.

Answer: C

The chain rule differentiates a composition of functions:

\[ \frac{d}{dx}f\big(g(x)\big) = f'\big(g(x)\big) \cdot g'(x) \]

The product rule handles a product of functions, \( (uv)' = u'v + uv' \), which is a different situation entirely.

Answer: A

An ordinary differential equation (ODE) involves derivatives with respect to a single independent variable.

\[ \frac{d^{2}y}{dx^{2}} + 3\frac{dy}{dx} + 2y = 0 \]

When two or more independent variables appear and partial derivatives are involved, the equation is a PDE instead.

Answer: C

Operational calculus replaces the operations of calculus with algebraic ones. The Laplace transform is its best-known tool: differentiation in \( t \) becomes multiplication by \( s \).

\[ \mathcal{L}\{f'(t)\} = sF(s) - f(0) \]

A differential equation therefore turns into an algebraic one, which is solved and then transformed back.