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Advance Engineering Mathematics : Question Paper May 2016 - Mechanical Engineering (Semester 3) | Gujarat Technological University (GTU)

## Advance Engineering Mathematics - May 2016

### Mechanical Engineering (Semester 3)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

### Answer the following one mark each questions

1(a) Integreating factor of the differential equation
$\dfrac{dx}{dy}+\dfrac{3x}{y}=\dfrac{1}{y^2}$ is _______
(1 marks)
1(b) The general solution of the differential equation $\dfrac{dy}{dx}+\dfrac{y}{x}=\tan 2x$ _______.(1 marks) 1(c) The orthogonal trajectory of the family of curve x2 + y2 = c2 is _______ .(1 marks) 1(d) Particular integral of (D2 + 4)y = cos 2x is _______ .(1 marks) 1(e) X=0 is a regular singular point of
$2x^2y''+3xy'(x^2-4)y=0\ \text{say true or false}$
(1 marks)
1(f) The solution of
(y ' z)p + (z ' x)q = x ' y is _______
(1 marks)
1(g) State the type ,order and degree of differential equation $\left ( \dfrac{dx}{dy} \right )^2+5y^{\dfrac{1}{3}}=x$ is _______(1 marks) 1(h) Solve (D+D')z=cos x(1 marks) 1(i) Is the partial differential equation $$2\dfrac{\partial ^2 u}{\partial x^2}+4\dfrac{\partial ^2 u}{\partial x \partial y}+3\dfrac{\partial ^2 u}{\partial y^2}=6\ \text {elliptic}?$$(1 marks) 1(j) $L^{-1}\left ( \dfrac{1}{(s+a)^2} \right )=$ _______(1 marks) 1(k) If f(t) is a periodic function with period t L [f(t)] = _______ .(1 marks) 1(l) Laplace transform of f(t) is defined for +ve and 've values of t. Say true or false.(1 marks) 1(m) State Duplication (Legendre) formula.(1 marks) 1(n) Find $B\left ( \dfrac{9}{2},\dfrac{7}{2} \right )$(1 marks) 2(a) Solve : 9y y' + 4x = 0(3 marks) 2(b) Solve : $\dfrac{dy}{dx}+y \cot x=2 \cos x$(4 marks)

### Solve any one question from Q.2(c) & Q.2(d)

2(c) Find series solution of y'' + xy = 0(7 marks) 2(d) Determine the value of $(a)J\frac{1}{2}(x)\ \ \ \ (b)J\frac{3}{2}(x)$(7 marks)

### Solve any three question from Q.3(a), Q.3(b), Q.3(c) & Q.3(d), Q.3(e), Q.3(f)

3(a) Solve (D2 + 9)y = 2sin 3x + cos 3x(3 marks) 3(b) Solve y'' + 4y' = 8x2 by the method of undetermined coefficients.(4 marks) 3(c) (i) Solve x2p + y2q = z2
(ii) Solve by charpit's method px+qy = pq
(7 marks)
3(d) Solve y'' + 4y' + 4 = 0 , y(0) = 1 , y'(0) = 1(3 marks) 3(e) Find the solution of y'' + a2y' = tan ax , by the method of variation of parameters.(4 marks) 3(f) Solve the equation ux = 2ut + u given u(x,0)=4e-4x by the method of seperation of variable.(7 marks)

### Solve any three question from Q.4(a), Q.4(b), Q.4(c) & Q.4(d), Q.4(e), Q.4(f)

4(a) Find the fourier transform of the function f(x) = e-ax2(3 marks) 4(b) Obtain fourier series to represent f(x) =x2 in the interval

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