HW3 - 293 HW 3 Geoffrey Recktenwald 9-13-2005 1.4-2 Solve...

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: 293 HW 3 Geoffrey Recktenwald 9-13-2005 1.4-2 Solve the ODE dy dx =- 2 xy 2 [Method - Separate]- dy y 2 = 2 xdx Z- dy y 2 = Z (2 xdx ) 1 y = x 2 + c y = 1 x 2 + c 1.4-7 Solve the ODE dy dx = (64 xy ) 1 / 3 dy dx = 4 3 x 3 y [Method - Separate] dy 3 y = 4 3 xdx Z dy 3 y = Z 4 3 xdx 3 2 y 2 / 3 = 4 3 4 x 4 / 3 + c y = 2 x 4 / 3 + C 3 / 2 1.4-25 Solve the ODE with IC x dy dx- y = 2 x 2 y dy dx = 2 x 2 y + y x [Method - Separate] dy y = 2 x 2 + 1 x dx Z dy y = Z 2 x + 1 x dx ln | y | = x 2 + ln | x | + c y = Cxe x 2 [Apply Initial condition y (1) = 1] 1 = Ce 1 2 C = e- 1 y = xe x 2- 1 1.4-40 Generic exponential decay equation A ( t ) = A e- kt For our example 1 2 = e- 5 . 27 k Solving for k yields k = ln(2) / 5 . 27 . 1315 The generic equation becomes A ( t ) = A e- . 1315 t 100 = 1 e- . 1315 t t = ln(100) / . 1315 35 . 01 The area is inhabitable in about 35 years. 1.4-50 Tripling every 7.5 years implies exponential growth. The Generic exponential growth equation is A ( t ) = A e kt For our example 3 = e 7 . 5 k Solve for k k = ln(3) / 7 . 5 = 2 15 ln(3) = ln(3 2 / 15 ) Our growth equation becomes A ( t ) = A e ln(3 2 / 15 ) t = A 3 2 t/ 15 a). The initial amount A = 10 so our equation becomes A ( t ) = 10 3 2 t/ 15 1 b). Given A = 10 how much ( A ( t )) will be present in 5 years? A ( t ) = 10 3 2(5) / 15 = 10 3 2 / 3 20 . 80 pu c). How long ( t ) will it take to reach A ( t ) = 100 ? 100 = 10 3 2 t/ 15 t = 15 2 ln 10 ln 3 15 . 72 years 1.4-57 Torricellis law Equation (1.4-24) A ( y ) dy dx =- k y For a Cylinder, A(y) is a constant so we absorb it into k dy dt =- K...
View Full Document

This note was uploaded on 04/01/2008 for the course MATH 2930 taught by Professor Terrell,r during the Fall '07 term at Cornell University (Engineering School).

Page1 / 5

HW3 - 293 HW 3 Geoffrey Recktenwald 9-13-2005 1.4-2 Solve...

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