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 u_x = 2u_t + 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
$ -\pi\lt/span\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(c)\lt/b\gt Find Half-Range cosine series for $$F(x)=\begin{matrix} kx & ,0\leq x\leq \dfrac{1}{2}\ k(l-x) & ,\dfrac{l}{2}\leq x\leq l \end{matrix}$$
Also prove that $ \sum {^\infty_{n=1}}=\dfrac{1}{(2n-1)^2}=\dfrac{\pi^2}{8} $(7 marks) 4(d) Expres the function $$F(x)=\begin{matrix} 2 & ,|x|<2\\ 0 & ,|x|>2\\ \end{matrix}\ \ \ \text{as Fourier integral}.$$(3 marks) 4(e) Find the fourier series expansion of the function $$F(x)=\begin{matrix} -\pi & -\pi\ltx\lt0\\ x="" &="" 0\ltx\lt\pi\\="" \end{matrix}$$<="" a="">

</x<0\\></span>(4 marks) 4(f) Find fourier series to represent the function
F(x) = 2x-x² in 0 < x < 3(7 marks)

Solve any three question from Q.5(a), Q.5(b), Q.5(c) & Q.5(d), Q.5(e), Q.5(f)

5(a) Find $ L{-1}\left \{ \dfrac{1}{(s+\sqrt{2})(s-\sqrt{3})} \right \} $(3 marks) 5(b) Find the laplace transform of $$(i)\dfrac{\cos at-\cos bt}{t}$$ $$(ii)t\sin at$$(4 marks) 5(c) State convolution theorem and use to it evaluate $$L^{-1}\left \{ \dfrac{1}{(s^2+a^2)^2} \right \}$$(7 marks) 5(d) $ (a)L\left \{ t^2 \cos h 3t\right \} $(3 marks) 5(e) Find $ L^{-1}\left \{ \dfrac{1}{s^4-81} \right \} $(4 marks) 5(f) Solve the equation y'' ' 3y' + 2y = 4t + e^3t , when y(0)=1 , y'(0) = -1(7 marks)