Partial Differential Equations MCQs

1

Advanced Mathematics • Partial Differential Equations

The steady-state solution mu(x,t) = mu_s(x) of the heat equation mu_t = a^2 mu_xx satisfying the conditions mu(0,t) = -1 and mu(2,t) = 1 is: The steady-state solution mu(x,t) = mu_s(x) of the heat equation mu_t = a^2 mu_xx satisfying the conditions mu(0,t) = -1 and mu(2,t) = 1 is:

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A. mu_s(x) = 1 - x mu_s(x) = 1 - x B. mu_s(x) = -1 + x mu_s(x) = -1 + x C. mu_s(x) = -1 + 3x mu_s(x) = -1 + 3x D. mu_s(x) = 1 + 2x mu_s(x) = 1 + 2x

सही उत्तर: B Correct Answer: B

At steady state, mu_xx = 0, so mu_s(x) = Ax + B. The boundary values give B = -1 and A = 1; hence mu_s(x) = -1 + x.

2

Advanced Mathematics • Partial Differential Equations

The equation of the characteristic curve of the one-parameter family (x - a)^2 + (y - a)^2 + z^2 = 1 is: The equation of the characteristic curve of the one-parameter family (x - a)^2 + (y - a)^2 + z^2 = 1 is:

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A. (x - y)^2 = 2(1 - z^2) (x - y)^2 = 2(1 - z^2) B. (x - y)^2 = 1 - z^2 (x - y)^2 = 1 - z^2 C. (x - z)^2 = 2(1 - y^2) (x - z)^2 = 2(1 - y^2) D. (x - z)^2 = 1 - y^2 (x - z)^2 = 1 - y^2

सही उत्तर: A Correct Answer: A

Differentiating with respect to the parameter gives a = (x + y)/2. Eliminating a from the family yields (x - y)^2 = 2(1 - z^2).

3

Advanced Mathematics • Partial Differential Equations

The partial differential equation obtained by eliminating the arbitrary function f from z = e^(mx)f(x + y) is: The partial differential equation obtained by eliminating the arbitrary function f from z = e^(mx)f(x + y) is:

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A. p - q = mz p - q = mz B. q - r = z q - r = z C. p - t = z p - t = z D. p + q = mz p + q = mz

सही उत्तर: A Correct Answer: A

Differentiating gives p = mz + e^(mx)f'(x+y) and q = e^(mx)f'(x+y). Hence p - q = mz.

4

Advanced Mathematics • Partial Differential Equations

The integral surface of yp + xq - z = 0 passing through the curve z = x^3, y = 0 is: The integral surface of yp + xq - z = 0 passing through the curve z = x^3, y = 0 is:

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A. z(1 + x^2 - y^2) = (x^2 - y^2)(x + y + z) z(1 + x^2 - y^2) = (x^2 - y^2)(x + y + z) B. 1 + x^2 - y^2 = (x^2 - y^2)(x + y + z) 1 + x^2 - y^2 = (x^2 - y^2)(x + y + z) C. z(1 - x^2 - y^2) = (x^2 - y^2)(x + y + z) z(1 - x^2 - y^2) = (x^2 - y^2)(x + y + z) D. z(1 + x^2 - y^2) = (x^2 + y^2)(x + y + z) z(1 + x^2 - y^2) = (x^2 + y^2)(x + y + z)

सही उत्तर: A Correct Answer: A

Solving the Lagrange first-order PDE by its characteristic equations and imposing z = x^3 on y = 0 yields the surface in option A.

5

Advanced Mathematics • Partial Differential Equations

The particular integral of the equation (2D^2 - D')z = 10e^(x - 3y) is: The particular integral of the equation (2D^2 - D')z = 10e^(x - 3y) is:

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A. 2e^(x - 3y) 2e^(x - 3y) B. e^(x - 3y) e^(x - 3y) C. 2e^(x + 3y) 2e^(x + 3y) D. e^(x + 3y) e^(x + 3y)

सही उत्तर: A Correct Answer: A

For e^(ax+by), replace D by a and D' by b. Here a = 1 and b = -3, so the operator value is 2(1)^2 - (-3) = 5. Thus the particular integral is (10/5)e^(x-3y) = 2e^(x-3y).

6

Advanced Mathematics • Partial Differential Equations

The integral curves of the equations dx/[x^2(y^2-z^2)] = dy/[y^2(z^2-x^2)] = dz/[z^2(x^2-y^2)] are given by, where c1 and c2 are arbitrary constants: The integral curves of the equations dx/[x^2(y^2-z^2)] = dy/[y^2(z^2-x^2)] = dz/[z^2(x^2-y^2)] are given by, where c1 and c2 are arbitrary constants:

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A. x^2 + y^2 - z^2 = c1 and x^-1 + y^-1 - z^-1 = c2 x^2 + y^2 - z^2 = c1 and x^-1 + y^-1 - z^-1 = c2 B. x^2 - y^2 + z^2 = c1 and x^-1 - y^-1 + z^-1 = c2 x^2 - y^2 + z^2 = c1 and x^-1 - y^-1 + z^-1 = c2 C. x^2 - y^2 - z^2 = c1 and x^-1 - y^-1 - z^-1 = c2 x^2 - y^2 - z^2 = c1 and x^-1 - y^-1 - z^-1 = c2 D. x^2 + y^2 + z^2 = c1 and x^-1 + y^-1 + z^-1 = c2 x^2 + y^2 + z^2 = c1 and x^-1 + y^-1 + z^-1 = c2

सही उत्तर: D Correct Answer: D

Using suitable multipliers in the Lagrange auxiliary system yields the two independent first integrals x^2+y^2+z^2=c1 and 1/x+1/y+1/z=c2.

7

Advanced Mathematics • Partial Differential Equations

Two ends A and B of a rod of length 20 cm have temperatures 40 degrees C and 90 degrees C until steady state. After steady state, the temperatures at A and B are changed to 45 degrees C and 95 degrees C respectively. In the stated temperature-distribution formula mu(x,t)=Ax+B+series terms, the values of A and B are: Two ends A and B of a rod of length 20 cm have temperatures 40 degrees C and 90 degrees C until steady state. After steady state, the temperatures at A and B are changed to 45 degrees C and 95 degrees C respectively. In the stated temperature-distribution formula mu(x,t)=Ax+B+series terms, the values of A and B are:

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A. A=3/2, B=42 A=3/2, B=42 B. A=45, B=5/2 A=45, B=5/2 C. A=5/2, B=3/2 A=5/2, B=3/2 D. A=5/2, B=45 A=5/2, B=45

सही उत्तर: D Correct Answer: D

The new steady component is linear with mu(0)=45 and mu(20)=95. Hence B=45 and A=(95-45)/20=5/2.

Partial Differential Equations MCQs

Partial Differential Equations MCQs

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