Applied Mathematics 2 - May 2015

First Year Engineering (Semester 2)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) $$ Evaluate \ \int^\infty_0 \dfrac {x^4}{4^x} dx $$(3 marks) 1 (b) Find P.I. of (D²-4D+4) y=e^x+cos 2x(3 marks) 1 (c) Show that ∇=1-E^-1.(3 marks) 1 (d) $$ Evaluate \ \int^1_0 \int^{\sqrt{1+x^2}}_0 \dfrac {dydx}{1+x^2+y^2} $$(3 marks) 1 (e) $$ Solve \ \left ( 1+e^{x/y} \right )dx+e^{x/y} \left ( 1- \dfrac {x}{y} \right )dy = 0 $$(4 marks) 1 (f) Evaluate $$ \int^\infty_0 \int^\infty_0 e^{-(x^2+y^2)}dxdy $$ by changing to polar co-ordinates(4 marks) 2 (a) Solve $$ y^4 dx = \left ( x^{-3/4} - y^{3}x \right )dy $$(6 marks) 2 (b) Change the order of integration and evaluate $$ \int^1_0 \int^{1/x}_0 \dfrac {y}{(1+xy)^2 (1+y^2)}dydx $$(6 marks) 2 (c) (i) $$ P.T. \ \int^\infty_0 \dfrac {x^{n=1}}{(a+bx)^{m-n}}dx = \dfrac {1}{a^n b^m} \beta (m,n) $$(4 marks) 2 (c) (ii) $$ P.T. \ \int^\infty_0 \dfrac{\log (1+ax^2)}{x^2} dx = \pi \sqrt{a}, \ where \ a>0 $$(4 marks) 3 (a) Evaluate $$ \int^{\log 2}_0 \int^x_0 \int^{x+\log y}_0 e^{x+y+z}dzdydx $$(6 marks) 3 (b) Find the area bounded between the paraboala
x²=4ay and x²=-4a(y-2a)(6 marks) 3 (c) Solve by the method of variation of parameters $$ \dfrac {d^2 y}{dx^2} + y = \sec x \tan x $$(8 marks) 4 (a) Find the length of the cardioid r=a(1-cos ?) lying outside the circle r=a cos ?(6 marks) 4 (b) Solve $$ \dfrac {d^2y}{dx^2} - 4 \dfrac {dy}{dx}+3y = 2xe^{3x}+3e^x \cos 2x $$(6 marks) 4 (c) Using R.K. Method of fourth order, solve, $$ \dfrac {dy}{dx} = \dfrac {y^2 - x^2}{y^2+x^2} \ given \ y(0)=1 \ at \ x=0.2, 0.4 $$(8 marks) 5 (a) Solve x sin x dy+(xycos x- ysinx -2)dx=0(6 marks) 5 (b) Solve $$ \dfrac {dy}{dx} =2 + \sqrt{xy} \ with \ x_0=1.2, y_0 =1.6403 $$ by moditied Euler's method, for x=1.4 correct to 4-decimal places, (taking h=0.2)(6 marks) 5 (c) Evaluate $$ \int^6_0 x f(x) dx \ by $$ i) Trapezoidal rule
ii) simpson's 1/3^rd rule
using the following table