VTU Engineering Maths 2 - December 2015 Exam Question Paper

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=e^2x-3sin 2x.

6 M

1 (b) By the method undetermined coefficients solve

\frac{d^{2} y}{d x^{2}} + y = 2 \cos x .

$\dfrac {d^2y}{dx^2}+ y = 2 \cos x.$

7 M

1 (c) Solve by the method of variation of parameter y''+4y=tan 2x.

7 M

2 (a) Solve

\frac{d^{6} y}{d x^{4}} + m^{4} y = 0.

$\dfrac {d^6y}{dx^4}+ m^4 y=0.$

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

\frac{d x}{d t} + 2 y + s i n t = 0, \frac{d y}{d t} - 2 x - \cos t = 0

$\dfrac {dx}{dt}+ 2y + sin t=0, \ \ \ \dfrac {dy}{dt}-2x - \cos t=0$ 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

\frac{d y}{d x} - \frac{d x}{d y} = \frac{x}{y} - \frac{y}{x} .

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

6 M

4 (a) Solve (x+a)² y''-4(x+a)y'+6y=x.

7 M

4 (b) Solve

P = \tan (x - \frac{p}{1 + p^{2}}) .

$P=\tan \left ( x - \dfrac {p}{1+p^2} \right ) .$

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

\int_{0}^{\infty} \int_{0}^{\infty} e^{- (x^{2} + y^{2})} d x d y

$\displaystyle \int^\infty_{0}\int^\infty_0 e^{-(x^2 + y^2)} dx \ dy$ by changing into polar co-ordinates.

6 M

6 (a) Solve

\frac{\partial^{1} Z}{\partial x \partial y} = \sin x \sin y, for which \frac{\partial z}{\partial y} = - 2 \sin y

$\dfrac {\partial^1 Z}{\partial x \partial y}= \sin x \sin y, \text { for which } \dfrac {\partial z}{\partial y}=-2 \sin y$ when x=0 and z=0, when y is an odd multiple of π/2.

7 M

6 (b) Evaluate

\iint_{R} x y d x d y,

$\displaystyle \iint_R \ xydxdy,$ where R is the region bounded by x-axis, the ordinate x=2a and the parabola x²=4 ay.

7 M

6 (c) Evaluate

\int_{- c}^{c} \int_{- b}^{b} \int_{- a}^{a} (x^{2} + y^{2} + z^{2}) d z d y d x .

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

6 M

7 (a) Define Gamma function Beta function. Prove that

Γ (1 / 2) \sqrt{π}

$\Gamma(1/2) \sqrt{\pi}$

7 M

7 (b) Express the vector

\bar{F} = z \hat{i} - 2 x \hat{j} + y \hat{k}

$\overline{F}=z\widehat{i}-2x\widehat{j}+y\widehat{k}$ 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

β (m, n) = \frac{Γ m Γ n}{Γ (m + n)}

$\beta (m, n) = \dfrac {\Gamma m \Gamma n}{\Gamma{(m+n)}}$

7 M

8 (b) Show that the area between the parabolas y²=4ax and x²=4ay is

\frac{16}{3}

$\dfrac {16}{3}$ a².

6 M

8 (c) Prove that the cylindrical co-ordinate system is orthogonal.

7 M

9 (a) Find L{e^-2t sin 3t+e^t t cost}.

7 M

9 (b) Find the inverse Laplace transform of

\frac{4 s + 5}{(s - 1)^{2} (s + 2)}

$\dfrac {4s+5}{(s-1)^2(s+2)}$ .

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 {\frac{\cos a t - \cos b t}{t}}

$L \left \{ \dfrac {\cos at - \cos bt}{t} \right \}$ .

7 M

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