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Partial Differential Equation MCQ - 2 - Mathematics MCQ


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15 Questions MCQ Test Topic-wise Tests & Solved Examples for Mathematics - Partial Differential Equation MCQ - 2

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Partial Differential Equation MCQ - 2 - Question 1


find the value of fy at (x, y) = (0, 1).

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 1

Using Euler theorem 
xfx + yfy = n f(x, y) 
Substituting x = 0; n = -96 and y = 1 we have 
fy = -96. f(0, 1) = -96.(1⁄1) 
= – 96.

Partial Differential Equation MCQ - 2 - Question 2

Find the derivative of Tan(x) = Tan(y).

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 2

Tan(y) = Tan(x)

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Partial Differential Equation MCQ - 2 - Question 3

General solution of pde given below is (y2+ z2+ x2 )p - 2xyq + 2xz = 0

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 3

ANSWER :- a

Solution :- Lagrange auxiliary equations are :-

dx/(y^2 + z^2 - x^2) = -dy/(-2xy) = -dz/(2xz)

Taking the last two members, we get

dy/y = dz/z

Integrating log y = logz + logc1

=y/z = c1

Using ,multipliers x,y,z we get (xdx + ydy + zdz)/-x(x^2 + y^2 + z^2) 

(xdx + ydy + zdz)/-x(x^2 + y^2 + z^2) = dx/(-2xz)

2(xdx + ydy + zdz)/(x^2 + y^2 + z^2) = dz/z

Integrating, log(x^2 + y^2 + z^2) = logz + logc2

(x^2 + y^2 + z^2) = zc2

Hence the required solution is f(c1,c2) = 0

= f(y/z, (x^2 + y^2 + z^2)/z) = 0

Partial Differential Equation MCQ - 2 - Question 4

If z = 3xy + 4x2, what is the value of ∂z / ∂x?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 4

Given: z = 3xy + 4x2 
Using partial differentiation, we need to differentiate the function z with respect to x keeping y as constant. Thus, ∂z / ∂x = 3y + 8x.

Partial Differential Equation MCQ - 2 - Question 5

The general integral of z(xp — yq) = y2 — x2 is

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 5

ANSWER :- c

Solution :-  z(xzx−yzy)=y^2−x^2

xzx−yzy = (y^2−x^2)z

dx/dt=x  , letting x(0)=1 , we have x=e^t

dy/dt=−y , letting y(0)=y0 , we have y=y0e^(−t) = y0/x

dz/dt=(y^2−x^2)/z

=y^20e^(−2t) −e^(2t)/z , 

we have z^2=f(y0)−y^20e^(−2t)−e^(2t) 

=f(xy)−y^2−x^2 ,

 i.e. x2+y2+z2=f(xy)

z^2=- x^2 - y^2 + f(xy)

Partial Differential Equation MCQ - 2 - Question 6

What is the derivative of with respect x?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 6

Partial Differential Equation MCQ - 2 - Question 7

What is the reason behind the non-existence of any real function which satisfies the differential equation, (y’)2 + 1 = 0?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 7

Given: (y’)2 + 1 = 0
Consider if y = 2x, then y’ = 2 and hence the left-hand side of the equation becomes 3 which is greater than 1. Therefore, the left-hand side of the equation will always be greater than, or equal to one and thus cannot be zero and hence the differential equation is not satisfied.

Partial Differential Equation MCQ - 2 - Question 8

Find 

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 8

We have

Partial Differential Equation MCQ - 2 - Question 9

What are the tangents to the curve x3 + y3 = 3axy at the origin?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 9

Given: x3 + y3 = 3axy
To find the tangent to the curve at the origin, we need to equate the lowest degree term to 0.
Therefore, 3axy = 0, which gives x = 0 and y = 0 as two tangents to the curve at origin.

Partial Differential Equation MCQ - 2 - Question 10

Let f : R2 → R  be defined by

Which of the following statements holds regarding the continuity and the existence of partial derivatives of f at (0, 0)?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 10

By the definition of partial derivatives


If we moving along the curve y = mx2

⇒ f(x, y)  is not continuous.

Partial Differential Equation MCQ - 2 - Question 11

Find the order of the difference equation Δ3yn – Δ2yn – Δyn = 3

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 11

Explanation: The first step is to expand the given equation by replacing every Δyn by (yn+1 – yn). Order of a difference equation is given by, (n+3-n)/1 which is actually 3.

Partial Differential Equation MCQ - 2 - Question 12

Find the differentiation of x3 + y3 – 3xy + y2 = 0?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 12

Differentiation of x3 is 3x2 
differentiation of y3 is 3y2 dy / dx
differentiation of -3xy is [-3y-3x(dy/dx)]
differentiation of y2 is 2y dy / dx
Hence, 

Partial Differential Equation MCQ - 2 - Question 13

Which of the following relations hold true for division rule of differentiation?

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 13

The division rule of differentiation for two functions is given by,

Partial Differential Equation MCQ - 2 - Question 14

The integral surface of the p. d. e satisfying the condition u ( 1 , y ) = y is given by

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 14

Think of the PDE as ∇u⋅(a(x,y),b(x,y))=0∇u⋅(a(x,y),b(x,y))=0 so that, geometrically it must be that dy/dx = [b(x,y)/a(x,y)] The solution of the ODE defines the characteristic curves of the PDE, along which solutions (to the PDE) are constant. Then u(x,y) = f(C) where the function ff is determined from the given auxiliary condition.

For example, here the ODE is dy/dx = y/x

 so C = ln|y/x| 

Then u(x,y) = f(C) = f(ln|y/x|).

To determine f, apply the given condition: u(1,y) = f(ln|y|) = y

⟹ f(y) = ey 

Thus, u(x,y) = exp(ln|y/x|)

= |y/x|

 

Partial Differential Equation MCQ - 2 - Question 15

When solved by the method of Differentiation for the given integral i.e the result obtained is given by _______.

Detailed Solution for Partial Differential Equation MCQ - 2 - Question 15

To solve this problem let us assume the given function is dependent on α Such that α = 2 & thus 

f (α) = log(α + 1) + c
or f (α) = log(α + 1) …… neglecting constant since the function is assumed
thus f (2) = log(2 + 1) = log(3).

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