2.06.MoreTools

# 2.06.MoreTools - :[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ Accumulation Publisher Math...

This preview shows pages 1–2. Sign up to view the full content.

Accumulation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 2.06 More Tools and Measurements: Techniques for Calculating Integrals BASICS B.1) Separating the variables and integrating to get formulas for solutions of some differential equations · B.1.a) Here is Mathematica 's formula for the solution of the differential equation y' @ x D = x H y @ x D + 4 L 2 with y @ 3 D = - 9. Clear @ y, x, sol D ; dsol = DSolve A9 y' @ x D == x H y @ x D + 4 L 2 , y @ 3 D == - 9 = , y @ x D , x E :: y @ x D Ø - 2 I - 81 + 10 x 2 M - 43 + 5 x 2 >> Use the technique of separating and integrating to explain where this formula comes from. · Answer: Rewrite y' @ x D = x H y @ x D + 4 L 2 to get y' @ x D H y @ x D + 4 L 2 = x. Some folks call this technique of putting all the y @ x D terms on one side and the x terms on the other side by the name "separation of variables." Integrate both sides from 3 to x to get Ÿ 3 x y' @ t D H y @ t D + 4 L 2 t = Ÿ 3 x t t , remembering that y @ 3 D = - 9: Clear @ x, y, t D ; left = 3 x y' @ t D H y @ t D + 4 L 2 t ê . y @ 3 D Ø - 9 - 1 5 - 1 4 + y @ x D This is Mathematica 's way of telling you that Ÿ 3 x y' @ t D H y @ t D + 4 L 2 t with y @ 3 D = - 9 is: betterleft = 1 13 - 1 4 + y @ x D 1 13 - 1 4 + y @ x D Now integrate the right hand side: right = 3 x t t - 9 2 + x 2 2 Here comes the formula for y @ x D : Solve @ betterleft == right, y @ x DD :: y @ x D Ø - 2 I - 225 + 26 x 2 M - 119 + 13 x 2 >> Compare: dsol :: y @ x D Ø - 2 I - 81 + 10 x 2 M - 43 + 5 x 2 >> Not bad, eh? · B.1.b) Here is Mathematica 's formula for the solution of the logistic differential equation y' @ x D = a y @ x D J 1 - y @ x D b N with y @ 0 D = 5. Clear @ y, x, a, b, c, sol D ; dsol = DSolve B: y' @ x D == a y @ x D 1 - y @ x D b , y @ 0 D == 5 > , y @ x D , x F :: y @ x D Ø 5 b a x - 5 + b + 5 a x >> Use the technique of separating and integrating to explain where this formula comes from. · Answer: Rewrite y' @ x D = a y @ x D J 1 - y @ x D b N by separating the variables: y' @ x D y @ x D I 1 - y @ x D b M = a. Integrate both sides from 0 to x to get Ÿ 0 x y' @ t D y @ t D I 1 - y @ t D b M t = Ÿ 0 x a t, remembering that y @ 0 D = 5: Clear @ x, y, t D ; left = 0 x y' @ t D y @ t D I 1 - y @ t D b M t ê . y @ 0 D Ø 5 - Log @ 5 D + Log @ 5 - b D + Log @ y @ x DD - Log @ - b + y @ x DD This is Mathematica 's way of telling you that Ÿ 0 x y' @ t D y @ t D I 1 - y @ t D b M t with y @ 0 D = 5 is - Log @ 5 D + Log @ 5 - b D + Log @ y @ x DD - Log @ - b + y @ x DD : Put this in better form, remembering that Log @ p D + Log @ q D = Log @ p q D and - Log @ r D = Log A 1 r E : Clear @ p, q, r, s D ; logrules = : Log @ p _ D + Log @ q _ D Ø Log @ p q D , - Log @ r _ D Ø Log B 1 r F> ; betterleft = left êê . logrules - Log @ 5 D + Log B H 5 - b L y @ x D - b + y @ x D F Now integrate the right hand side: right = 0 x a t a x Now set both sides equal to each other and solve for

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/11/2011 for the course MATH 231 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

### Page1 / 19

2.06.MoreTools - :[email protected] Accumulation Publisher Math...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online