Sol-HW2 - Math Methods Solutions for Assignment 2 Sep. 01,...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math Methods Solutions for Assignment 2 Sep. 01, 2010 Fall 2010 Differential Equations Due Sep. 08, 2010 1 (6). Find the general solutions of the following equations: a) (4 y- 3 x ) dx + ( y- 2 x ) dy = 0. Since the coefficients are homogenous functions of order one, we use u = y/x . 4 y x- 3 dx + y x- 2 dy = 0 (4 u- 3) dx + ( u- 2)( udx + xdu ) = 0 ( u 2 + 2 u- 3) dx + x ( u- 2) du = 0 dx x + u- 2 ( u + 3)( u- 1) du = 0 dx x + 5 / 4 u + 3- 1 / 4 u- 1 du = 0 ln( x ) + 5 4 ln( u + 3)- 1 4 ln( u- 1) = c x 4 ( u + 3) 5 u- 1 = c x 4 ( y/x + 3) 5 y/x- 1 = c ( y + 3 x ) 5 y- x = c b) y (4) + 2 y 00 + y = 0. The substitution y = e x yields 4 + 2 2 + 1 = ( - i ) 2 ( + i ) 2 = 0 . Linearly independent solutions are e ix ,xe ix ,e- ix and xe- ix . In terms of functions of real variables these are sin x,x sin x, cos x and x cos x . y = c 1 cos x + c 2 sin x + c 3 x cos x + c 4 x sin x c) y 00 + 1 x y + 1 x 2 y = x . This is an Euler equation. First, we search for solution of homogenous equation as y = x . ( - 1) + + 1 = 0; 2 + 1 = 0; = i. Linearly independent solutions are x i = e i ln x and x- i = e- i ln x , and the general solution of homogenous equation is y = c 1 cos(ln x ) + c 2 sin(ln x ). Since the inhomogenous part of the equation is a polynomial, it is easier to search for the particular solution as a polynomial. We can search for it as y = cx 3 , substituting we get 6 cx + 3 cx + cx = x , i.e. c = 1 / 10. Hence y = c 1 cos(ln x ) + c 2 sin(ln x ) + 1 10 x 3 2 (4). Find the general solutions of the equation (1- 2 x ) y 00 + 4 xy- 4 y...
View Full Document

This note was uploaded on 09/28/2011 for the course PHYSICS 801 taught by Professor Ivanov during the Fall '10 term at Kansas State University.

Page1 / 5

Sol-HW2 - Math Methods Solutions for Assignment 2 Sep. 01,...

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