When using the variable separable method to solve a partial differential equation, then the function can be written as the product of functions depending only on one variable. For example, U(x,t) = X(x)T(t).

While solving a partial differential equation using a variable separable method, we assume that the function can be written as the product of two functions which depend on one variable only.

A

True

B

False

If u (x, y, z, t) = f(x + iβy - vt) + g(x - iβy - vt), where f and g are arbitrary and twice differentiable functions, is a solution of the wave equation $$\frac{{\partial {u^2}}}{{\partial {x^2}}} = \frac{{{\partial ^2}u}}{{\partial {y^2}}} = \frac{1}{{{c^2}}}\frac{{{\partial ^2}u}}{{\partial {t^2}}}$$ then β is

A

$${\left( {1 - \frac{v}{c}} \right)^{\frac{1}{2}}}$$

B

$$\left( {1 - \frac{v}{c}} \right)$$

C

$${\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)^{\frac{1}{2}}}$$

D

$$\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)$$

While solving a partial differential equation using a variable separable method, we equate the ratio to a constant which?

A

can be positive or negative integer or zero

B

can be positive or negative rational number or zero

C

must be a positive integer

D

must be a negative integer

The figure shows the plot of y as a function of x
Differential Equations mcq question image

The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is

A

$$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 1$$

B

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{x}}$$

C

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - {\text{x}}$$

D

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$

The equation relating E, P, V and T which is true for all substanes under all conditions is given by $${\left( {\frac{{\partial {\text{E}}}}{{\partial {\text{V}}}}} \right)_{\text{T}}} = {\text{T}} \cdot {\left( {\frac{{\partial {\text{P}}}}{{\partial {\text{T}}}}} \right)_{\text{H}}} - {\text{P}}{\text{.}}$$ This equation is called the

A

Maxwell's equation

B

Thermodynamic equation of state

C

Equation of state

D

Redlich-Kwong equation of state

Maxwell equation $${\left( {\frac{{\partial P}}{{\partial T}}} \right)_V} = {\left( {\frac{{\partial S}}{{\partial V}}} \right)_T}$$ is produced from which of the following equation

A

dU = Tds - PdV

B

dH = Tds + VdP

C

dA = - PdV - SdT

D

dG = VdP - SdT

In a certain code language, O is written as R, R is written as A, A is written as I, I is written as N, N is written as E, E is written as P, P is written as T, C is written as M and L is written as L. How will PORCELAIN be written as in that language?

A

ERAMPLINT

B

TAMRPLINE

C

PLINETRAM

D

TRAMPLINE

While solving any partial differentiation equation using a variable separable method which is of order 1 or 2, we use the formula of fourier series to find the coefficients at last.

A

True

B

False

In a certain code language, W is written as B, K is written as H, L is written as U, N is written as A, O is written as M, E is written as R, D is written as G, G is written as E, then how will KNOWLEDGE be written in that code?

A

MAHBERGUR

B

HAMBURGER

C

HAMBUGRED

D

MAHBURGER

As per the RBI Act, 1934, the following functions are described as the functions of a Central Bank:
1. Banking functions
2. Advisory functions
3. Supervisory functions
4. Promotional functions
Identify the correct combination:

A

1, 3 and 4

B

1, 2 and 4

C

2, 3 and 4

D

Only 1 and 3

When using the variable separable method to solve a partial differential equation, then the function can be written as the product of functions depending only on one variable. For example, U(x,t) = X(x)T(t).

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