Y',y'',y''' ... notation is termed as
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.
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When ∊ is a constant than ∂/∂x(∊x)=
Answer: C
A constant factor passes straight through differentiation.
\[ \frac{\partial}{\partial x}(\epsilon x) = \epsilon \cdot \frac{\partial x}{\partial x} = \epsilon \cdot 1 = \epsilon \]
Since \( \epsilon \) does not depend on \( x \), it is not itself differentiated.
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Separation of variables method is also termed as
Answer: B
Separation of variables is also called the Fourier method, after Fourier's use of it on the heat equation.
The idea is to look for a solution built as a product of single-variable functions:
\[ u(x,t) \;=\; X(x)\,T(t) \]
Substituting and dividing through leaves each side depending on one variable only, so both must equal a constant. That turns one partial differential equation into two ordinary ones, whose solutions are then recombined as a Fourier series.
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A differential equation is said to be separable if variables can be
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.
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Model rests on neither theory nor observation, but is merely invocation of expected structure is
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.
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A parametric curve that represents a specific solution to an ordinary differential equation or system of equations is called
Answer: A
An integral curve is a curve whose tangent at every point matches the direction the differential equation prescribes there - that is, a particular solution drawn in the plane.
The family of all such curves fills the direction field, and an initial condition picks out one of them.
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If in interval a and b are finite and are included, interval is called
Answer: B
An interval is closed when both endpoints are finite and both belong to it, written with square brackets:
\[ [a,b] \;=\; \{\, x : a \le x \le b \,\} \]
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If number has one digit in integral part, then its characteristic is
Answer: D
The characteristic is one less than the count of digits before the decimal point.
\[ \text{characteristic} = 1 - 1 = 0 \]
A one-digit number lies between 1 and 10, so its logarithm lies between \( \log 1 = 0 \) and \( \log 10 = 1 \) - the whole part must therefore be 0. For instance \( \log 7 = 0.8451 \).
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Siny=1 is
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.
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Distance from Earth to Sun is approx
Answer: A
The mean Earth–Sun distance is about \( 1.5 \times 10^{8} \) km - 150,000,000 km.
This distance is itself the definition of the astronomical unit (AU), the standard yardstick for distances within the solar system. Light covers it in about 8 minutes 20 seconds.
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