Engineering Mathematics 4 - Dec 2012

Computer Science Engg. (Semester 4)

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) Using the Taylor's series method, solve the initial value problem $$ \dfrac {dy}{dx}=x^2 y-1, y(0)=1 $$ at the point x=0.1(6 marks) 1 (b) Employ the fourth order Runge-Kutta method to solve $$ \dfrac {dy}{dx}= \dfrac {y^2-x^2}{y^2+x^2} y(0)=1 $$ at the points x=0.2 and x=0.4. Take h=0.2.(7 marks) 1 (c) $$ Given \ \dfrac {dy}{dx}= xy+y^2, \ y(0)=1, \ y(0.1)=1.1169, \ y(0.2)=1.2773, \ y(0.3)=1.5049 $$ Find y(0.4) using the Milne's predictor-corrector method, Apply the corrector formula twice.(7 marks) 2 (a) Employing the Picard's method, obtain the second order approximate solution of the following problem at x=0.2. $$ \dfrac {dy}{dx}=x+yz, \ \dfrac {dz}{dx}= y+zx, \ y(0)=1, \ z(0)=-1 $$(6 marks) 2 (b) Using the Runge-kutta method, find the solution at x=0.1 of the differential equation $$ \dfrac {d^2y}{dx^3}- x^2 \dfrac {dy}{dx}- 2xy =1$$ under the conditions y(0)=1, y'(0)=0. Take step length h=0.1.(7 marks) 2 (c) Using the Milne's method, obtain an approximate solution at the point x=0.4 of the problem $$ \dfrac {d^2y}{dx^2}+3x \dfrac {dy}{dx}-6y=0, \ y(0)=1, \ y'(0)=0.1 $$ Given y(0.1)=1.03995, y'(0.1)=0.6955, y(0.2)=1.138036, y'(0.2)=1.258, y(0.3)=1.29865, y'(0.3)=1.873.(7 marks) 3 (a) If f(z)=u+iv is an analytic function then prove that $$ \left [ \dfrac {\partial }{\partial x} |f(z)| \right ]^2 + \left [ \dfrac {\partial }{\partial y} |f(z)| \right ]^2 =|f'(z)|^2 $$(6 marks) 3 (b) Find an analytic function whose imaginary part is v=e^x{(x²-y²) cos y-2xy sin y}(7 marks) 3 (c) If f(z)=u(r,?)+ iv(r,?) is an analytic function, show that u and v satisfy the equation $$ \dfrac {\partial^2 \varphi}{\partial r^2}+ \dfrac {1}{r} \dfrac {\partial \varphi}{ \partial r}+ \dfrac {1}{r^2} \dfrac {\partial^2 \varphi}{\partial \theta^2}=0 $$(7 marks) 4 (a) Find the bilinear transformation that maps the points 1, i, -1 onto the point i, 0, -i respectively.(6 marks) 4 (b) Discuss the transformation W=e^x.(7 marks) 4 (c) Evaluate $$ \int_C \dfrac {\sin \pi z^2 + \cos \pi z^3}{(z-1)^2 (z-2)}dz $$ where C is the circle |z|=3.(7 marks) 5 (a) Express the polynomial 2x³-x²-3x+2 in terms of Legendre polynomials.(6 marks) 5 (b) Obtain the series solution of Bessel's differential equation $$ x^2 \dfrac {d^2 y}{dx^2}+ x \dfrac {dy}{dx}+ (x^2-n^2)y=0 $$ in the form y=AJ_n(x) + BJ_-n(x).(7 marks) 5 (c) Derive Rodrique's formula $$ P_n(x)= \dfrac {1}{2^n n!} \dfrac {d^n}{dx^n}(x^2-1)^n. $$(7 marks) 6 (a) State the axioms of probability. For any two events A and B, Prove that P(A?B)=P(A)+P(B)-P(A?B).(6 marks) 6 (b) A bag contains 10 white balls and 3 red ball while another bag contains 3 white balls and 5 red balls. Two balls are drawn at random from the first bag and put in the second bag and then a ball is drawn at random from the second bag. What is the probability that it is a white ball?(7 marks) 6 (c) In a bolt factory there are four machines A, B, C, D manufacturing respectively 20%, 15%, 25%, 40% of the total production. Out of these 5%, 4%, 3% and 2% respectively are defective. A bolt is drawn at random from the production and is found to be defective. Find the probability that it was manufactured by A or D.(7 marks) 7 (a) The probability distribution of finite random variable x is given by the following table: