Download PTU B-Tech CE 2020 Dec 2nd Sem 76254 Mathematics Ii Question Paper

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Roll No. Total No. of Pages : 02

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Total No. of Questions : 18

B.Tech (Civil Engg.) (2018 & Onwards) (Sem.-2)

MATHEMATICS-II

Subject Code : BTAM-201-18

M.Code : 76254

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Time : 3 Hrs. Max. Marks : 60

INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION -B & C have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.

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4. Select atleast TWO questions from SECTION - B & C.

SECTION-A

Answer briefly :

1. Is this differential equation $z''x - \frac{2}{y} + a^2x = 0$ linear?
2. Is this differential equation $x^2 ydx - (x^3 + y^3) dy = 0$ exact?

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3. Write the solution of the Clairaut’s equation $y= px + sin^{-1} p$.
4. Find the wronskian from $\frac{d^2y}{dx^2} + 4y = \tan 2x$.
5. Find complementary function of $\frac{\partial^2 z}{\partial x^2} - 2\frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} = \sin x$.
6. Find particular integral of $(D_x^2 - a^2) Z = x^2 \sin pt$.
7. Write one dimensional wave equation.

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8. Classify the equation $(x + 1) u_{xx} - 2(x + 2)u_{xy} + (x + 3) u_{yy} = 0$.
9. What is a boundary value problem?
10. Write Laplace equation in cylindrical coordinates.

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SECTION-B

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1. Solve a) $[1+\log(xy)]dx+\{1+\frac{x}{y}\}dy =0$. b) $x[\frac{dy}{dx}+y]=1-y$.
2. a) Solve $(D^2 - 6D + 9)y=6e^{3x}+7e^{-2x} - \log 2$. b) Find the power series solution of the differential equation $(4x^2 D^2+ 2D + 1) y = 0$.
3. Solve a) $p\sqrt{x} +q\sqrt{y} =z$ b) $xp+yq=z(x+y)$.
4. a) Solve the PDE $(D^2 - 2DD’ + D'^2)z =e^{x+y}$. b) Solve the PDE $(D + D') (D — 2D’ +2) z = \sin (2x + y)$.

SECTION-C

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1. Solve $4\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} =3u$ by method of separation of variables. Given that $u = 3e^{-y} - e^{-5x}$ when $x = 0$.
2. Solve the BVP $\frac{\partial^2 u}{\partial t^2} =c^2 \frac{\partial^2 u}{\partial x^2}$ using D’ “Alembert’s technique subject to the conditions $u=P_y \cos pt$ when $x =l$ and $u = 0$ when $x =0$.
3. Solve the BVP $\frac{\partial^2 u}{\partial t^2} =c^2 \frac{\partial^2 u}{\partial x^2}$ using separation of variables method subject to the conditions $u (0, t) = u (l, t) =0$, $u (x, 0) =x$ where $l > 0$.
4. The diameter of a semi-circular plate of radius a is kept at 0°C and the temperature at the semicircular boundary is T°C. Estimate the steady state temperature in the plate using the Laplace equation $r^2\frac{\partial^2 u}{\partial r^2} + r\frac{\partial u}{\partial r} + \frac{\partial^2 u}{\partial \theta^2} =0$.

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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