Advanced Engineering Mathematics - May 2016

Civil Engineering (Semester 3)

TOTAL MARKS:
TOTAL TIME: HOURS

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)