Engineering Maths 2 - Dec 2015

First Year Engineering (P Cycle) (Semester 2)

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. 1 (a) Solve y''+4y'-12y=e^2x-3sin 2x.(6 marks) 1 (b) By the method undetermined coefficients solve $ \dfrac {d^2y}{dx^2}+ y = 2 \cos x. $(7 marks) 1 (c) Solve by the method of variation of parameter y''+4y=tan 2x.(7 marks) 10 (a) Express $ f(t) = \begin{bmatrix} \cos t &0 2\pi \end{bmatrix} $ in terms of unit step function and hence find its Laplace transform.(7 marks) 10 (b) Solve by Laplace transform y''+6y'+9y=12t² e^-3t with y(0)=0=y'(0).(6 marks) 10 (c) Find $ L \left \{ \dfrac {\cos at - \cos bt}{t} \right \} $.(7 marks) 2 (a) Solve $ \dfrac {d^6y}{dx^4}+ m^4 y=0. $(6 marks) 2 (b) Solve (D²+7D+12)y=cos hx.(7 marks) 2 (c) By the method variation of parameters, solve y''+y=x sin x.(7 marks) 3 (a) Solve the simultaneous equations $$ \dfrac {dx}{dt}+ 2y + sin t=0, \ \ \ \dfrac {dy}{dt}-2x - \cos t=0 $$ given that x=0 and y=1 when t=0.(7 marks) 3 (b) Solve x² y''-xy'+2y=x sin (log x).(7 marks) 3 (c) Solve $ \dfrac {dy}{dx} - \dfrac {dx}{dy} = \dfrac {x}{y} - \dfrac {y}{x} . $(6 marks) 4 (a) Solve (x+a)² y''-4(x+a)y'+6y=x.(7 marks) 4 (b) Solve $ P=\tan \left ( x - \dfrac {p}{1+p^2} \right ) . $(7 marks) 4 (c) Find the general and the singular solution of the equation y=px+p³.(6 marks) 5 (a) Form the Partial Differential Equation of z=y f(x)+x g(y), where f and g are arbitrary functions.(7 marks) 5 (b) Derive one dimensional heat equation.(7 marks) 5 (c) Evaluate $ \displaystyle \int^\infty_{0}\int^\infty_0 e^{-(x^2 + y^2)} dx \ dy $ by changing into polar co-ordinates.(6 marks) 6 (a) Solve $ \dfrac {\partial^1 Z}{\partial x \partial y}= \sin x \sin y, \text { for which } \dfrac {\partial z}{\partial y}=-2 \sin y $ when x=0 and z=0, when y is an odd multiple of π/2.(7 marks) 6 (b) Evaluate $ \displaystyle \iint_R \ xydxdy, $ where R is the region bounded by x-axis, the ordinate x=2a and the parabola x²=4 ay.(7 marks) 6 (c) Evaluate $ \displaystyle \int^c_{-c} \int^b_{-b} \int^a_{-a} (x^2+y^2 + z^2) \ dz \ dy \ dx. $(6 marks) 7 (a) Define Gamma function Beta function. Prove that $ \Gamma(1/2) \sqrt{\pi} $(7 marks) 7 (b) Express the vector $ \overline{F}=z\widehat{i}-2x\widehat{j}+y\widehat{k} $ in cylindrical co-ordinates.(6 marks) 7 (c) Find the volume common on the cylinders x²+y²=a² and x²+z²=a².(7 marks) 8 (a) Prove that $ \beta (m, n) = \dfrac {\Gamma m \Gamma n}{\Gamma{(m+n)}} $(7 marks) 8 (b) Show that the area between the parabolas y²=4ax and x²=4ay is $ \dfrac {16}{3} $a².(6 marks) 8 (c) Prove that the cylindrical co-ordinate system is orthogonal.(7 marks) 9 (a) Find L{e^-2t sin 3t+e^t t cost}.(7 marks) 9 (b) Find the inverse Laplace transform of $ \dfrac {4s+5}{(s-1)^2(s+2)} $.(6 marks) 9 (c) Solve y''-6y' + 9y=12t² e^-3t by Laplace transform method with y(0)=0=y(0).(7 marks)