## Mathematics 2 - Dec 2014

### First Year Engineering (Semester 2)

TOTAL MARKS:

TOTAL TIME: HOURS

**1** Find a particular integral of the differential equation (D^{3}+6D+5)y=e.(2 marks)
**10** State final value theorem.(2 marks)

### Answer any one question from Q11 (a) & Q11 (b)

**11 (a) (i)** Solve the differential equation y''+a^{2}y=tan ax by variation of parameters method.(8 marks)
**11 (a) (ii)** Solve the following simultaneous differential equations. $$ \dfrac {dx}{dt} + 2x -3y = t \ and \ \dfrac {dy}{dt}- 3x + 2y =e^{2t} $$(8 marks)
**11 (b) (i)** Solve ((x+1)^{2}D^{2}+(x+1)D+1)y=4 cos log (x+1).(8 marks)
**11 (b) (ii)** Solve (D^{2}-7D-6)y=(1+x)e^{2x}.(8 marks)

### Answer any one question from Q12 (a) & Q12 (b)

**12 (a) (i)** Prove that $$(\phi \bar{F})= \phi div \bar{F}+ \nabla \phi\cdot \bar{F} $$ Also determine the value of n for which $$ r^{n}\bar{R}\ is \ solenoidal \ where \ \bar{R} = x\bar{i}+ y\bar{j}+ z\bar{k} \ and \ r=|\bar{R}|. $$(8 marks)
**12 (a) (ii)** Verify Gauss divergence theorem for $$ \bar{F}= x^2 \overrightarrow{i}+ y^2 \overrightarrow{j} + z^2 \overrightarrow{k} $$ over the volume of the cuboid formed by the planes x=0, x=a, y=0, y=b, z=0 and z=c.(8 marks)
**12 (b) (i)** Prove that $$ \bar{F} = (y^2 + 2xz^2 ) \bar{i} + (2xy - z) \bar{j}+ (2x^2 z - y+2z) \bar{k} $$ irrotational and hence find its scalar potential.(8 marks)
**12 (b) (ii)** Verify Stokes' theorem for $$ \bar{F}= (x^2 +y^2) \bar{i} + 2xy \bar{j} $$ where S is the rectangle in the xy-plane formed by the lines x=0, x=a, y=0 and y=b.(8 marks)

### Answer any one question from Q13 (a) & Q13 (b)

**13 (a) (i)** If f=u+iv is an analytic function, prove that $$ \left ( \dfrac {\partial ^2 } {\partial x^2 }+ \dfrac {\partial^2}{\partial y^2} \right )| f(z)|^2=4 |f'(z)|^2 $$(8 marks)
**13 (a) (ii)** Find the bilinear transformation which maps the points ∞, 2, -1 to 1, ∞ and 0 respectively.(8 marks)
**13 (b) (i)** Find the analytic function f=u+iv given that. u(x,y)=e^{2x} (x sin 2y+ y cos 2 y).(8 marks)
**13 (b) (ii)** If f=u+iv is analytic on a domain D and |f| is constant on D, prove that f must be a constant on D.(8 marks)

### Answer any one question from Q14 (a) & Q14 (b)

**14 (a) (i)** $$ If \ F(a)= \oint_c \dfrac {3z^2+7z+1}{z-a}dz $$ where C:|z|=2 and |a|≠2, find F and F''(1-i).(8 marks)
**14 (a) (ii)** Evaluate $$ \int^\infty_0 \dfrac {x^2}{(x^2+a^2) (x^2 +b^2)}dx \ $$ by the method of contour integration, if a and b are positive.(8 marks)
**14 (b) (i)** Find the Laurent's series of $$ f(z)= \dfrac {3z-2}{z(z^2-4)} $$ valid in the region 2<|z+2|<4.(8 marks)
**14 (b) (ii)** Using contour integration method show that $$ \int^{2x}_0 \dfrac {d\theta}{a+b \cos \theta}= \dfrac {2\pi}{\sqrt{a^2 -b^2}} $$ if a>b>0.(8 marks)

### Answer any one question from Q15 (a) & Q15 (b)

**15 (a) (i) ** Find the Laplace transform of $$ f(t) = \dfrac {\sin^2 t}{t} $$(4 marks)
**15 (a) (i) ** Find the value of $$ \int^\infty_0 te^{3t} \cos 2tdt $$(4 marks)
**15 (a) (ii)** Solve y''+9y=cos 2t given that y(0)=1 and y(π/2)=-1, by given by the method of Laplace transform.(8 marks)
**15 (b) (i) ** Find $$ L^{-1} \left ( \log \dfrac {s^2 +1} { s (s+1)} \right ) $$(4 marks)
**15 (b) (i) ** Using convolution theorem, find y if $$ L(y)= \dfrac {s}{(s^2 +a^2)^2} $$(4 marks)
**15 (b) (ii)** Find L(f(t)) if $$f(t) = \begin{cases}t & 0\le t\le a\\2a-t & a\le t\le 2a\end{cases}$$ and $$f(t+2a)= f(t)$$(8 marks)
**2** Transform the differential equation x^{2}y''-xy'+2y=0 with constant coefficients.(2 marks)
**3** Find $$\nabla (\nabla . ((x^2 - yz) \bar{i} + (y^2 -xz)\bar{j} + (z^2 -xy) \bar{k})) $$ at the point (1, -1, 2).(2 marks)
**4** State Green's theorem in the plane.(2 marks)
**5** Give an example of a complex-valued function which is differentiable at a point but not analytic at that point.(2 marks)
**6** $$ If \ u(x,y)= 3x^2 y + 2x^2 - y^3 - 2y^2, $$ verify whether u is harmonic.(2 marks)
**7** State Cauchy's integral formula.(2 marks)
**8** Find the residue of $$ \left \{ \dfrac {\sin 3z}{z^6} \right \} \ at \ z=0 $$(2 marks)
**9** State sufficient conditions for the existence of Laplace transform.(2 marks)