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VTU First Year Engineering (P Cycle) (Semester 2)
Engineering Maths 2
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

1 (a) Solve y''+4y'-12y=e2x-3sin 2x.
6 M
1 (b) By the method undetermined coefficients solve $$\dfrac {d^2y}{dx^2}+ y = 2 \cos x.$$
7 M
1 (c) Solve by the method of variation of parameter y''+4y=tan 2x.
7 M

2 (a) Solve $$\dfrac {d^6y}{dx^4}+ m^4 y=0.$$
6 M
2 (b) Solve (D2+7D+12)y=cos hx.
7 M
2 (c) By the method variation of parameters, solve y''+y=x sin x.
7 M

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 M
3 (b) Solve x2 y''-xy'+2y=x sin (log x).
7 M
3 (c) Solve $$\dfrac {dy}{dx} - \dfrac {dx}{dy} = \dfrac {x}{y} - \dfrac {y}{x} .$$
6 M

4 (a) Solve (x+a)2 y''-4(x+a)y'+6y=x.
7 M
4 (b) Solve $$P=\tan \left ( x - \dfrac {p}{1+p^2} \right ) .$$
7 M
4 (c) Find the general and the singular solution of the equation y=px+p3.
6 M

5 (a) Form the Partial Differential Equation of z=y f(x)+x g(y), where f and g are arbitrary functions.
7 M
5 (b) Derive one dimensional heat equation.
7 M
5 (c) Evaluate $$\displaystyle \int^\infty_{0}\int^\infty_0 e^{-(x^2 + y^2)} dx \ dy$$ by changing into polar co-ordinates.
6 M

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 M
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 M
6 (c) Evaluate $$\displaystyle \int^c_{-c} \int^b_{-b} \int^a_{-a} (x^2+y^2 + z^2) \ dz \ dy \ dx.$$
6 M

7 (a) Define Gamma function Beta function. Prove that $$\Gamma(1/2) \sqrt{\pi}$$
7 M
7 (b) Express the vector $$\overline{F}=z\widehat{i}-2x\widehat{j}+y\widehat{k}$$ in cylindrical co-ordinates.
6 M
7 (c) Find the volume common on the cylinders x2+y2=a2 and x2+z2=a2.
7 M

8 (a) Prove that $$\beta (m, n) = \dfrac {\Gamma m \Gamma n}{\Gamma{(m+n)}}$$
7 M
8 (b) Show that the area between the parabolas y2=4ax and x2=4ay is $$\dfrac {16}{3}$$a2.
6 M
8 (c) Prove that the cylindrical co-ordinate system is orthogonal.
7 M

9 (a) Find L{e-2t sin 3t+et t cost}.
7 M
9 (b) Find the inverse Laplace transform of $$\dfrac {4s+5}{(s-1)^2(s+2)}$$.
6 M
9 (c) Solve y''-6y' + 9y=12t2 e-3t by Laplace transform method with y(0)=0=y(0).
7 M

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 M
10 (b) Solve by Laplace transform y''+6y'+9y=12t2 e-3t with y(0)=0=y'(0).
6 M
10 (c) Find $$L \left \{ \dfrac {\cos at - \cos bt}{t} \right \}$$.
7 M

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