GTU Advanced Engineering Maths - December 2015 Exam Question Paper

GTU Information Technology (Semester 3)
Advanced Engineering Maths
December 2015

Total marks: --
Total time: --

INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

Answer the following one mark each questions:

1(a) Find

$\Gamma \left ( \dfrac{13}{2} \right )$

1 M

1(b) State relationship between beta and gamma functions.

1 M

1(c) Represent graphically the given saw-tooth function f(x) = 2x, 0 ≤ x < 2 and f(x + 2) = f(x) for all x.

1 M

1(d) For a periodic function f with fundamental period p, state the formula to find Laplace transform of f.

1 M

1(e) Find

$L\left ( e^{-3t}f(t) \right )$ , if

$L\left ( f(t) \right )=\dfrac{s}{(s-3)^2}$ .

1 M

1(f) Find

$L[(2t-1)^2]$ .

1 M

1(g) Find the extension of the function f(x) = x + 1, define over (0,1] to [?1, 1] - {0} which is an odd function.

1 M

1(h) Is the function \[f(x)=\left\{\begin{matrix} x & 0\leq x\leq 2\\ x^2 & 2

1 M

1(i) Is the differential equation

$\dfrac{dy}{dx}=\dfrac{y}{x}$ exact? Give reason.

1 M

1(j) Give the differential equation of the orthogonal trajectory to the equation y = cx² .

1 M

1(k) If y = c ₁ y₁+c₂ y ₂ = e^x (c₁ cos x + c ₂ sin x) is a complementary function of a second order differential equation, find the Wronskian W(y₁ , y₂ ).

1 M

1(l) Solve (D² + D + 1)y = 0; where

$D=\frac{\mathrm{d} }{\mathrm{d} t}$ .

1 M

1(m) Is

$u(t,x)=50e^{(t-x)/2}$ , a solution to

$\dfrac{\partial u}{\partial t}=\dfrac{\partial u}{\partial x}+u$ ?

1 M

1(n) Give an example of a first order partial differential equation of Clairaut's Form.

1 M

2(a) Solve:

$\dfrac{dy}{dx}=\dfrac{x^2-x-y^2}{2xy}$ .

3 M

2(b) Solve

$\dfrac{dy}{dx}+\dfrac{1}{x}y=x^3y^3$ .

4 M

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

2(c) Find the series solution of

$(x-2)\dfrac{d^2y}{dx^2}-x^2\dfrac{dy}{dx}+9y=0$ about x₀ = 0.

7 M

2(d) Explain regular-singular point of a second order differential equation and find the roots of the indicial equation to x²y''+xy'-(2-x)y=0.

7 M

Solved any one question from Q.3 & Q.4

3(a) Find the complete solution of

$\dfrac{d^3y}{dx^3}+8y=\cos h(2x)$ .

3 M

3(b) Find solution of

$\dfrac{d^2y}{dx^2}+9y=\tan 3x$ , using the method of variation of parameters.

4 M

3(c) Using separable variable technique find the acceptable general solution to the one-dimensional heat equation

$\dfrac{\partial u}{\partial t}=c^2\dfrac{\partial ^2 u}{\partial x^2}$ and find the solution satisfying the conditions u(0, t) = u(π , t) = 0 for t > 0 and u(x, 0) =π- x, 0 < x < π ..

7 M

4(a) Solve completely, the differential equation

$\dfrac{d^2y}{dx^2}-6\dfrac{dy}{dx}+9=\cos(2x)\sin x$ .

3 M

4(b) Solve completely the differential equation

$x^2\dfrac{d^2y}{dx^2}-6x\dfrac{dy}{dx}+6y=x^{-3}\log x$ .

4 M

4(c)i Form the partial differential equation for the equation (x-a)(y-b)-z²=x²+y².

3 M

4(c)ii Find the general solution to the partial differential equation xp+yp=x-y.

3 M

Solved any one question from Q.5 & Q.6

5(a) Find the Fourier cosine integral of

$f(x)=\dfrac{\pi}{2}e^{-x},x\geq 0$ .

3 M

5(b) For the function

$f(x)=\cos 2x$ , find its Fourier sine series over [0, π].

4 M

5(c) For the function

$f(x)=\left\{\begin{matrix} x & 0\leq x\leq 2\\ 4-x; & 2\leq x\leq 4 \end{matrix}\right.$ , find its Fourier series. Hence show that

$\dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots=\dfrac{\pi^2}{16}$ .

7 M

6(a) Find the Fourier cosine series of f(x)=e^-x, Where 0≤ x≤ π .

3 M

6(b) Find the Fourier cosine series of

$\int ^\infty _0 \dfrac{\lambda^3 \sin \lambda x}{\lambda^4+4}d\lambda=\dfrac{\pi}{2}e^{-x}\cos x,x>0$ .

4 M

6(c) Is the function f(x) = x+|x|, -π ≤ x ≤ π even or odd? Find its Fourier series over the interval mentioned.

7 M

Solved any one question from Q.7 & Q.8

7(a) Find

$L\left \{ \int ^t_0e^u (u+\sin u)du \right \}$ .

3 M

7(b) Find

$L^{-1}\left \{ \dfrac{1}{s(s^2-3s+3)} \right \}$ .

4 M

7(c) Solve the initial value problem:

$y''-2y'=e^t\sin t, y(0)=y'(0)=0$ using Laplace transform.

7 M

8(a) Find

$L\left \{ t(\sin t-t \cos t) \right \}$ .

3 M

8(b) Find

$L^{-1}\left \{ \dfrac{e^{-2s}}{(s^2+2)(s^2-3)} \right \}$ .

4 M

8(c) State the convolution theorem and verify it for f(t) = t and g(t) = e^2t.

7 M

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