2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
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1 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h arbirar consan C,,,. Eprimn wih diffrn inrvals for unil ou hav a plo ha shows wha ou considr o b h mos imporan bhavior of h famil. 0.5 () C () C 0.5 () () () PS.#4 Show ha () + C is a gnral soluion of h diffrnial quaion +. Us a compur or calculaor o skch h soluions for h givn valus of h arbirar consan C,,,. Eprimn wih diffrn inrvals for unil ou hav a plo ha shows wha ou considr o b h mos imporan bhavior of h famil. () + C () C () + () DIFFERENTIAL EQUATIONS Pag
2 MATH 5 PS # Summr 00 PS.#7 Plo h dircion fild for h diffrnial quaion + b hand. Do his b drawing shor lins of h appropria slop cnrd a ach of h ingr valud coordinas (,), whr and. Tabl and graph: PS.#0 Plo h dircion fild for h diffrnial quaion ( )/(+ ) b hand. Do his b drawing shor lins of h appropria slop cnrd a ach of h ingr valud coordinas (,), whr and. Tabl and graph: ( )/(+ ) PS.#4 Us a compur o draw a dircion fild for h givn firs-ordr diffrnial quaion (+)/( ), R{(,): 5 5 and 5 5}. Us h indicad bounds for our displa window. Obain a prinou and us pncil o draw a numbr of possibl soluion rajcoris on h dircion fild. If possibl, chck our soluions wih a compur. DIFFERENTIAL EQUATIONS Pag
3 MATH 5 PS # Summr 00 dp PS.#5 Bacria in a pri dish is growing according o h quaion 0.44P, whr P is d h mass of h accumulad bacria (masurd in milligrams) afr das. Suppos ha h iniial mass of h bacrial sampl is.5 mg. Us a numrical solvr o sima h amoun of bacria afr 0 das. Th iniial diffrnial quaion is P 0. 44P, P ( 0). 5 Bas on h graph, w sima ha P(0) 4. Thus, hr ar approimal 4 mg of baria prsn afr 0 das. DIFFERENTIAL EQUATIONS Pag
4 MATH 5 PS # Summr 00.. Soluions o Sparabl Equaions #, 6, 0, PS.# Find h gnral soluion of h diffrnial quaion plici soluion.. If possibl find an d d d d +C ()ln( +C) PS.#6 Find h gnral soluion of h diffrnial quaion +. If possibl find an plici soluion. d + d d ( - (-)+(-) ( +)(-) +)d d ( + ) d ln C -A A C + PS.#0 Find h gnral soluion of h diffrnial quaion. If possibl find an plici soluion. d + ( + ) d d + d d ( + ) d ln +ln +C + ln + C A PS.# Find ac soluions for h diffrnial quaion ( +)/, (). Sa h inrval of isnc. Plo ach ac soluion on h inrval of isnc. Us a numrical solvr o duplica h soluion curv for h iniial valu problm. d + d ln + + C Iniial condiion () d( + ) d d + + d C + + () ± A DIFFERENTIAL EQUATIONS Pag 4 A - ± A A A -4 A5 So, h paricular soluion is () 5 Th inrval of isnc: 5 > 0 5 > > > -ln5 5
5 MATH 5 PS # Summr 00 ln5 >- ln 5 (-,+ ) Plo of h ac soluion: Us numrical solvr o duplica h soluion curv: DIFFERENTIAL EQUATIONS Pag 5
6 MATH 5 PS # Summr Linar Equaions #4, 5, 6, 4, 9, 8, 9 PS.4#4 Find h gnral soluion of h firs-ordr, linar quaion +5. Mulipling boh sids b an ingraing facor, u() quaion bcoms +( ) 5 ( ) 5 5 d 5 5 d( ) + C 5 + C PS.4#5 Find h gnral soluion of h firs-ordr, linar quaion /(+)(+). ( + ) d Mulipling boh sids b an ingraing facor, u() +, h quaion bcoms ( + ) - (( + ) ) ( + ) ( + ) d + C d ( C + ) ( +) PS.4#6 Find h gnral soluion of h firs-ordr, linar quaion Rwri h diffrnial quaion as -4 Mulipling boh sids b an ingraing facor, u() h quaion bcoms ( ) 4 d ln + C 4 ln + C 4 PS.4#4 Find h soluion of h iniial valu problm + 4 d, (0). d, h ln + 4 ln 4 + 4, Rwri h diffrnial quaion as DIFFERENTIAL EQUATIONS Pag 6
7 MATH 5 PS # Summr 00 Mulipling boh sids b an ingraing facor, u() quaion bcoms - ( ) d d ( ) ( d) ( ) + C Iniial condiion (0)-+C C 5. So, h paricular soluion is () ( ) + 5 ( ) + C d, h PS.4#9 Find h soluion of h iniial valu problm (+) +(+) /, ( )0. Discuss h inrval of isnc and provid h skch of our soluion. u() + bcoms Rwri h diffrnial quaion as / ( + Mulipling boh sids b an ingraing facor, d ln + ( + / / - ( ( / (assuming +>0), h quaion / ( / (( ) ( / ( ( d ln( + C / ( / ln( + C( Iniial condiion 0(-)C. / ( So, h paricular soluion is () ln(. Th inrval of isnc: +>0 (,+ ). DIFFERENTIAL EQUATIONS Pag 7
8 MATH 5 PS # Summr 00 PS.4#8 Suppos ha ou hav a closd ssm conaining 000 individuals. A flu pidmic sars. L N() rprsn h numbr of infcd individuals in h closd ssm a im. Assum ha h ra a which h numbr of infcd individuals is changing is joinl proporional o h numbr of infcd individuals and o h numbr of uninfcd individuals. Furhrmor, suppos ha whn 00 individuals ar infcd, h ra a which individuals ar bcoming infcd is 90 individuals pr da. If 0 individuals ar infcd a im 0, whn will 90% of h populaion b infcd? Hin: Th assumpion hr is ha hr ar onl halh individuals and sick individuals. Furhrmor, h rsuling modl can b solvd using h chniqu inroducd in Ercis. Th modl quaion is N kn( 000 N). Whn N00, h ra of infcion is 90/da. So k k Thrfor, N N N 0.00 L N. Thn ( N N 0.00) / N 00 N / N Rwri h diffrnial quaion as N Mulipling boh sids b an ingraing facor, u(), h quaion bcoms ( ) d 0.00( + C) 0.00( + C ) d N () 000 /( + C ) Iniial condiion 0N(0) 000 /( + C) 50 So, N() 000 /( + 49 ). N() /( + 49 ) 900 C. + C ln(44) das. PS.4#9 In Ercis of Scion., h im of dah of a murdr vicim is drmind using Nwon s law of cooling. In paricular i was discovrd ha h proporionall consan in Nwon s law was kln(5/4) 0.. Suppos w discovr anohr murdr vicim a midnigh wih a bod mpraur of o C. Howvr, his im h air mpraur a midnigh is 0 o C, and is falling a a consan ra of o C pr hour. A wha im did h vicim di? (Rmmbr ha h normal bod mpraur is 7 o C.) Th modl quaion is T k( T + ). A midnigh T(0). Rwri h diffrnial quaion as DIFFERENTIAL EQUATIONS Pag 8
9 MATH 5 PS # Summr 00 T + kt k Mulipling boh sids b an ingraing facor, u() k, h quaion bcoms k k k T + k T k ( k k T ) k T () A midnigh T(0) So, T () A k T ( k k ) d ( k k k d ) ( d) + C k C k + k k kd k k + + C C. k k + ( ). k k ko o, w hav T ( o ) 7 o + ( ) k k ln(5 / 4) ln(5/ 4) 4 o 4.48 o ( ) 7 5 Th approimaion of o is Thus, h im of dah is approimal :PM. 7, whr o 5 k ln 4 ln(5 / 4) ( ) o + 7 DIFFERENTIAL EQUATIONS Pag 9
10 MATH 5 PS # Summr Miing Problms #, 4, 6 PS.5# A ank conains 00 gal of pur war. A im zro, a sugar-war soluion conaining 0.lb of sugar pr gal nrs h ank a a ra of gal pr minu. Simulanousl, a drain is opnd a h boom of h ank allowing h sugar soluion o lav h ank a gal pr minu. Assum ha h soluion in h ank is kp prfcl mid a all ims. (a) Wha will b h sugar conn in h ank afr 0 minus? (b) How long will i ak h sugar conn in h ank o rach 5lb? (c) Wha will b h vnual sugar conn in h ank? S() h amoun of sugar in h ank. Th ra in gal/min 0. lb/gal 0.6 lb/min. S Th ra ou gal/min lb/gal S lb/min. Hnc, ds ra in ra ou S d S S 0.6 Mulipling boh sids b an ingraing facor, u() quaion bcoms 0. 0 S + 0.0S ( S) S d 0 + C. 0 S () 0 + C 0. 0 Th iniial condiion 0 (0) So, S () S (0) d 0. 0 S 0 + C 0 C 0 (a) Th sugar conn in h ank afr 0 minus is 9.04 lb (b) S () (c) lim S( ) lim(0 0 0 ) ln , h min PS.5#4 A ank conains 500 gal of a sal-war soluion conaining 0.05 lb of sal pr gallon of war. Pur war is pourd ino h ank and a drain a h boom of h ank is adjusd so as o kp h volum of soluion in h ank consan. A wha ra (gal/min) should h war b pourd ino h ank o lowr h sal concnraion o 0.0 lb/gal of war in undr on hour? () h amoun of sal in h soluion a im. r h ra ha war nrs (and lavs) h ank. DIFFERENTIAL EQUATIONS Pag 0
11 MATH 5 PS # Summr 00 Considr sal in h ank: Th ra in 0 gal/min. () Th ra ou r gal/min 500 lb/gal 0.00r( ) lb/min. Hnc, d ra in ra ou 0.00r d d d 0.00r 0.00rd d 0. 00r+ C ln ( ) 0.00r + C () Th iniial condiion 0.05 (0) () A A 0. 00r So, () r. Th concnraion rachs % in 60 minus, so 0.r 0.0 (60) r ln(5) ln(5) r 0. 0.r 5.4 gal/min PS.5#6 A ank iniiall conains 00 gal of a sal-war soluion conaining 0.05 lb of sal for ach gallon of war. A im zro, pur war is pourd ino h ank a a ra of gal/min. Simulanousl, a drain is opnd a h boom of h ank ha allows h sal-war soluion o lav h ank a a ra of 4 gal/min. Wha will b h sal conn in h ank whn prcisl 50 gal of sal soluion rmain? Th volum of sal soluion h ank is dcrasing a (4-)gal/min. So a im, h volum of sal soluion in h ank is V ( ) 00. S () h amoun of sal in h ank. Th iniial condiion S ( 0) Th ra (Sal) in 0 Th ra (Sal) ou gal/min Hnc, d ds d S 00 ln S ( ) ln 00 + C () S( ) V ( ) ds ra in ra ou S S () ln 00 + C A( 00 ) lb/gal S 00 S( ) V ( ) S( ) 00 Th iniial condiion 5 S (0) 00 A 5 So, S () Whn ( ) (00 ). V, w hav So, h sal conn in h ank a ha im 50 lb. A 5 0 is (50) S DIFFERENTIAL EQUATIONS Pag
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