TIRUMALA ENGINEERING COLLEGE, BOGARAM.SUB: MATHEMATICAL METHODS

ASSIGNMENT: COMMON TO ALL BRANCHESI-B.TECH--A.Y-2009-2010

UNIT-VI

1. a. Find y at x = .1, .2 &.3 given that dy/dx = xy+y2 , y(0)1 by using Taylor’s series method.

b. Evaluate the values of y(1.1) & y(1.2) from y11+y2+y1 = x3 , y(1) = 1,y1(1) =1 by using Taylor’s series method.

2. Find y(.1) , y(.2) , z(.1) & z(.2) given that dy/dx = x+z , dz/dx = x-y2 & y(0) = 2, z(0) = 1 by using

Taylor’s series method.

3. a. Solve y1 = y-x2, y (0) = 1, by Picards method up to the 3rd approximation. Hence find the values of y(.1),y(.2)

b. Obtain y (.1) given y1 = (y-x)/(y+x) , y(0) = 1 by Picards method.

4. a. y1 = (x3+xy2)e-x , y(0) = 1 , y(.1), y(.2) by Euler’s method. b. Using Euler’s method, solve for y at x = 2 from y1=3x2+1 ,y(1) = 2 taking step size i) h=.5 ii) h= .25

5. Find y(.5),y(1) &y(1.5) given that y1 = 4-2x ,y(0) =2 with h = .5using Euler’s Modified method.

6. Given that y1 = x+siny, y (0) =1, compute y (.2) &y (.4) with h=.2 using Euler’s Modified method.

7. a. Find y(.1) & y(.2) using R-K-4th order formula given that y1=x2-y & y(0)=1.

b. Evaluate y (.8) using R-K-4th order method given y1=(x+y)1/2 , y = .41 at x = .48. a. Tabulate the values of y(.1), y(.2) & y(.3) using R-K-4th order formula given that y1 = xy+y2, y(0) = 1. b. Find the solution of y1=x-y at x=.4 such that y = 1 at x= 0 and h=.1 using Milne’s method. Use Euler’s Modified method to evaluate y (.1), y (.2) & y (.3).

9. a. y1 = 2ex-y , y(0)=2,y(.1)=2.010,y(.2)=2.040,y(.3)=2.09,find at x =.4&.5 by Milne’s Predictor Corrector Method.

b. Obtain y(.6)&y(.8) given y1 = x+y , y(0)=1 with h = .2 by Adam’s method.

10. a. Obtain the solution of y1 = x2(1+y), y(1) = 1 at x = 1.1,1.2 &1.3 by any numerical method and estimate at X = 1.4 by Adam’s method.

b. dy/dx = 2exy , y(0) = 2 , find y(.4) using Adams Predictor Corrector formula by calculate y(.1) , y(.2) & y (.3) using Euler’s Modified formula.

UNIT-VII

1. Find the Fourier series of period 2 π for the function f(x) = x2-x in (- π, π).

Hence deduce the sum of the series 1/12 +1/22 + 1/32+……. = π 2/6

2. Find the Fourier series of period 2 π for the function f(x) = - π , - π < x < 0

X , 0 < x < π.

Hence deduce the sum of the series 1/12 +1/32 + 1/52+……. = π 2/8

3. Obtain a Fourier expansion for

(1-cosx) 1/2 in - π < x < π

4. Find the Fourier expansion of f(x) = xcosx , 0 < x < 2 π

5.i. Expand f(x) = eax, 0 < x < 2 π

ii. Obtain the F.S. of f(x) = ( π –x )2 , 0 < x < 2 π

6. Find the sine series for f(x) = x( π – x ) , 0 < x < π

& hence deduce that 1/13 - 1/33 + 1/53 - 1/73 +……….. = π 2 /32

7. Find F.S. of xsinx as cosine series in (0, π) &

S.T. 1/1.3 - 1/3.5 + 1/ 5.7 – 1/ 7.9 +…… = (π -2) / 4

8.i. If F(x) = kx , 0 < x < π /2

K (π -x), π /2 < x < π, Find the half – range sine series.

ii. If F(t) = t , 0 < t < π / 2

π /2 , π /2 < t <. Π. Find Fourier sine series.

9. If f(x) = π x , 0 ≤ x ≤ 1 π (2-x) , 1 ≤ x ≤ 2.

Show that in (0, 2), f(x) = π / 2 - 4/ π {cosπx/12 +cos3πx /32 +cos5π x/52 +……}

& hence deduce that 1/12 +1/32+1/52+…… = π 2/8.

10. Find the Fourier series for f(x) = 2x-x2 , 0 < x < 3.

UNIT-VIII

1. Form the P.D.E. x2 + y2 + (z2-c2) = a2

2. i. . Form the P.D.E z = y f (x2+y2)

ii. Form the P.D.E xyz = f(x2+y2+z2)

3. Form the P.D.E. by eliminating arbitrary functions... z = f (2x+y) + g (3x-y)

4. Solve (x2-yz)p + (y2-zx)q = z2-xy

5. Solve (y+z) p + (z +x) q = x+y

6. Solve the p.d.e. x2p2 + y2q2 = 1

7. Solve the p.d.e. p2x + q2y = z

8. Solve the p.d.e. x2/p + y2/q = z

9. i. Solve by the method of separation of variables 2x ∂z / ∂x - 3y ∂z/∂y = 0

ii. Solve ∂2z / ∂x2 - 2 ∂z/∂x + ∂z/∂y = 0 by the method of separation of variables.

10. Solve 4 ∂u /∂x + ∂u /∂y = 3u & u = e-5y , where x = 0 .

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