2132Final_Practice_Answers

2132Final_Practice_Answers - Polytechnic Institute of NYU...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Polytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2008B Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet problems. These are additional practice problems designed to cover the material, but not necessarily specific to the exam. The final is cumulative, covering all material in the course. In general, 75-80% of the exam is new material. (1) Consider the initial value problem P 0 = te t 2 - P , y (0) = 1 . (a) Use the Euler Method with stepsize h = 0 . 25 to find the approximate value for the solution of the above problem at x = 0 . 5. We will need (0 . 5 - 0)( . 25) = 2 iterations to get the answer. n t n = t 0 + n * h y n f ( t n ,P n ) = te t 2 - P P n +1 = P n + h * f ( t n ,P n ) 0 0 0 0 * e 0 2 - 0 = 0 P 1 = 0 + (0 . 25)(0) = 0 1 0 . 25 0 0 . 25 * e (0 . 25) 2 - 0 = 0 . 2661 P 2 = 0 + 0 . 25(0 . 2661) = 0 . 0665 2 0 . 5 0 . 0665 n/a n/a (b) Find the exact solution and determine its value at x = 0 . 5. Separable: e P dP = te t 2 dt so P = ln(0 . 5 * e t 2 + C ) Using the IVP (0 , 0), 0 = ln(0 . 5 * 1 + C ) 1 = 0 . 5 + C C = 0 . 5 . So, P = ln(0 . 5 * e t 2 + 0 . 5) and P (0 . 5) = 0 . 1328. The approximation is poor due to the relatively large stepsize. (2) Consider the initial value problem x 2 y 0 = 2 xy - x 3 , y (1) = 3 . (a) Use the Euler Method with stepsize h = 0 . 1 to find the approximate value for the solution of the above problem at x = 1 . 3. n x n y n f ( x n ,y n ) = 2 * ( y/x ) - x y n +1 = y n + h * f ( x n ,y n ) 0 1 3 2 * 3 / 1 - 1 = 5 y 1 = 3 + (0 . 1)(5) = 3 . 5 1 1 + . 1 = 1 . 1 3 . 5 5 . 2636 y 2 = 3 . 5 + 0 . 1(5 . 2636) = 4 . 02636 2 1 . 2 4 . 02636 5 . 5106 y 3 = 4 . 02636 + 0 . 1(5 . 5106) = 4 . 577 3 1 . 3 4 . 577 n/a n/a 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 (b) Find the exact solution and determine its value at x = 1 . 3. First order linear: You must reformulate the problem as y 0 = 2 y/x - x as in the f ( x n ,y n ) column above. The integrating factor is e - R 2 /xdx = e - 2 ln | x | = x - 2 . Then x - 2 y 0 - 2 x - 3 y = - x - 1 . ( x - 2 y ) = - ln | x | + C y = - x 2 ln | x | + Cx 2 . Using the IVP (1 , 3), C = 3, so y = - x 2 ln | x | + 3 x 2 . Then y (1 . 3) = 5 . 5134 . (3) Consider the initial value problem xy 0 = y (ln( y/x ) + 1) , y (1) = e. (a) Use the Euler Method with stepsize h = 0 . 1 to find the approximate value for the solution of the above problem at x = 1 . 2. n x n y n f ( x n ,y n ) = ( y/x ) * (ln( y/x ) + 1) y n +1 = y n + h * f ( x n ,y n ) 0 1 e e * (ln( e ) + 1) = 2 e y 1 = e + 0 . 1 * (2 e ) = 1 . 2 * e = 3 . 2619 1 1 . 1 3 . 2619 6 . 1888 y 2 = 3 . 8808 2 1 . 2 3 . 8808 n/a n/a (b) Find the exact solution and determine its value at
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 8

2132Final_Practice_Answers - Polytechnic Institute of NYU...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online