# Solutions 1.4-Page 40

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1 Soluions.4-Pag 40 Problm Find gnral soluions (implici if ncssary, plici if convnin) of h diffrnial quaions. dy = d ( 4y) / Sparaing h variabls yilds: / / / y dy = 4 d y / dy = 4 / d Ingraing boh sids o obain / / y dy = 4 d / y 4 / = + / 4 / y = + y() yilds: 4 / y ( ) = ( + ) / Whr =

2 Problm 9 Find gnral soluions (implici if ncssary, plici if convnin) of h diffrnial quaions. dy ( ) = y d Sparaing h variabls yilds: dy d = y Ingraing boh sids o obain dy d = y can b dcomposd ino Th abov ingral bcoms dy d d = y + + ln y = + ) ) + y = y = + ) ) + ln ( + ) + y( ) = Whr = ) y() yilds: + +

3 Problm 5 Find plici paricular soluions of h iniial valu problm. dy y = y, y( ) = d Th variabls in h diffrnial quaion ar sparad as follows. dy = y( + ) d dy ( + ) d = = ( + )d y Ingraing boh sids o obain dy = ( + ) d y ln y = + ln + y = y = Whr = + ln + y() yilds: Th iniial condiion is usd o solv for h consan. y() = = = () y () bcoms y( ) = Simplifying givs ( y ( ) = )

4 Problm 9 (Populaion growh) A crain ciy had a populaion of 5000 in 90 and a populaion of 0000 in 90. Assum ha is populaion will coninu o grow ponnially a a consan ra. Wha populaion can is ciy plannrs pc in h yar 000? k From pg., P( ) = P0, whr is h im in yars. Assuming 90 is h iniial im, h givn informaion is as follows: P = P(0) = 0000 k is solvd for using h givn informaion as follows: k (0) P(0) = 0000 = 5000 k (0) = 5 5) k = 0 Now h populaion in h yar 000 or, P (40) = (40) 0 P( 40) = 5,840 popl = P(40), can b found.

5 Problm (Radiocarbon daing) arbon racd from an ancin skull conaind only on-sih as much 4 as carbon racd from prsn-day bon. How old is h skull? Th govrning diffrnial quaion is givn on pg.5 dn = kn d Th soluion of h diffrnial quaion is k N( ) N = 0 N I is givn ha a h currn im, N = 0. From pg.5, k = if is in yars. Th quaion o b solvd is: N 0 (0.000 = N ) 0 Dividing boh sids by (0.000) = = givs N 0 = 4,5 yars

6 Problm 9 A pichr of burmilk iniially a 5 is o b coold by sing i on h fron porch, whr h mpraur is 0. Suppos ha h mpraur of h burmilk has droppd o 5 afr 0 min. Whn will i b a 5. Th govrning diffrnial quaion is Eq.9 pg.. dt = k( A T ) d A is h mdium mpraur and im is masurd in minus for his problm. I is givn ha A is 0, so h diffrnial quaion bcoms dt = kt d Sparaing variabls yilds: dt = kd T Ingraing boh sids o g T () yilds: dt = kd T ln T = k + k+ T = k T = Whr = Th iniial condiion is givn as T ( 0) = 5. Subsiuing h iniial condiion ino h abov quaion givs T (0) = 5 = () = 5 T = 5 k I is also givn ha T ( 0) = 5. This informaion is usd o solv for k. T (0) = 5 = 5 5) k = 0 k (0) Th im whn h mpraur rachs 5 can b solvd for as follows:

7 T ( ) = 5 = = 5 5) 5) = 0 = min 5 0

8 Problm 44 According o on cosmological hory, hr wr qual amouns of h wo uranium isoops 5 U and 8 U a h craion of h univrs in h big bang. A prsn hr ar. aoms of 8 U for ach aom of 5 U. Using h half-livs yars for 8 U and yars for 5 U, calcula h ag of h univrs. M will dno h numbr of aoms of 5 U and N will dno h numbr of aoms of 8 U. Th form of h govrning quaion is N = N 0 k I can b shown ha h half-lif, τ, is ln τ = (s pg.) k Th consan, k, can b solvd for bcaus h half-livs ar givn. ln k = τ k =.540 k 8 5 = 9.0 N = N 0 M = M Th iniial im is h bginning of h univrs. Dividing boh quaions a h prsn im givs N ( ) =. = M ln. = ( ) = 5.99 billion yars

9 Problm 4 A crain pic of dubious informaion abou phnylhylamin in h drinking war bgan o sprad on day in a ciy wih a populaion of 00,000. Wihin a wk, 0000 popl had hard his rumor. Assum ha h ra of incras of h numbr who hav hard h rumor is proporional o h numbr who hav no y hard i. How long will i b unil half h populaion of h ciy has hard h rumor? An ODE mus b formd o fi h abov informaion. L N b h numbr of popl, in housands, who hav hard h rumor. Th ODE is: dn = k( 00 N) d Th abov ODE sas ha h ra of incras of h numbr who hav hard h rumor is proporional o h numbr who hav no y hard i, as sad in h problm samn. Tim is masurd in days. 00-N is h oal populaion minus h popl who hav hard h rumor, and hus h numbr of popl who hav no hard h rumor. Sparaing variabls yilds: dn = kd 00 N Ingraing boh sids o obain dn = kd 00 N 00 N) = k + 00 N = 00 N = Whr = ( k+ ) k N() yilds: Iniially, no on has hard h rumor, so N(0) = 0. This iniial condiion is usd o solv for = () = 00 N() = 0 is givn and is usd o solv for k. k () N() = 0 = = 00 k = k 9 0 )

10 Now h im whn half h populaion has hard h rumor can b found. 50 = = = 9 0 ) = Solving for givs: = 4 days

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