math 310 a4 Sol - MATH 310-3 Fall 2011 Differential...

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 310-3 Fall 2011 Differential Equations Homework Set 4 Some Solutions The following solutions are not necessarily complete, but should highlight the main points for most questions. If you have any further questions regarding the problems and solutions, feel free to discuss them on WebCT. Solutions marked by * are taken from the publishers Instructors Solutions Manual for the textbook by Boyce and DiPrima, 9th edition (Wiley), while solutions marked by (which appear to have been typed with a typewriter...) are from the publishers Student Solutions Manual; in some cases I have added comments in [brackets]. Section 2.2: * Section 2.2 # 14 : * Section 2.2 # 22 : * Section 2.2 # 24 : Section 2.2 # 30 : Following the steps outlined in the problem, we begin with dy dx = y- 4 x x- y = ( y/x )- 4 1- ( y/x ) . As indicated, the appropriate substitution for homogeneous equations is v = y/x . Hence y ( x ) = xv ( x ), so by the product rule, dy/dx = v + xdv/dx . Substituting, x dv dx + v = v- 4 1- v = x dv dx = v- 4 1- v- v = v 2- 4 1- v , which is a separable equation. Separating (for v 2 6 = 4), we need to integrate (1- v ) / ( v 2- 4), which we rewrite using partial fractions: 1- v v 2- 4 = 1- v ( v + 2)( v- 2) = A v + 2 + B v- 2 = ( A + B ) v + 2( B- A ) v 2- 4 , which is true for all v (comparing coefficients of powers of v ) if A + B =- 1 and 2( B- A ) = 1, so A =- 3 / 4, B =- 1 / 4. Returning to the differential equation for v ( x ), we have Z 1- v v 2- 4 dv =- Z 1 4 3 v + 2 + 1 v- 2 dv = Z 1 x dx + c = ln | x | + c, so 3ln | v + 2 | + ln | v- 2 | = ln | v + 2 | 3 | v- 2 | =- 4(ln | x | + c ) = ln | x |- 4- 4 c = | v + 2 | 3 | v- 2 | = C | x |- 4 (where C = e- 4 c > 0) ....
View Full Document

Page1 / 7

math 310 a4 Sol - MATH 310-3 Fall 2011 Differential...

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