Question Solved1 Answer 17) Extra Credit: The population P of elephants in a wild animal preserve changes with time (in years) according to the equation dP-0.0005P(190-P)dt = 0. Find P as a function of t if there are 46 elephants initially. 190 13.13e-0.095t 190 1 4.13e-0.095t 190 1 3.13e-0.95t A)P= B) P = 190 1+3.13e-0.95t C) P D) P=

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Transcribed Image Text: 17) Extra Credit: The population P of elephants in a wild animal preserve changes with time (in years) according to the equation dP-0.0005P(190-P)dt = 0. Find P as a function of t if there are 46 elephants initially. 190 13.13e-0.095t 190 1 4.13e-0.095t 190 1 3.13e-0.95t A)P= B) P = 190 1+3.13e-0.95t C) P D) P=
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Transcribed Image Text: 17) Extra Credit: The population P of elephants in a wild animal preserve changes with time (in years) according to the equation dP-0.0005P(190-P)dt = 0. Find P as a function of t if there are 46 elephants initially. 190 13.13e-0.095t 190 1 4.13e-0.095t 190 1 3.13e-0.95t A)P= B) P = 190 1+3.13e-0.95t C) P D) P=
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Given equation{:[dP-0.005 P(190-P)dt=0.],[dP=0.0005 P(190-P)dt],[(dP)/(dt)=(0.0005 xx190)P-0.0005P^(2)],[(dP)/(dt)-(0.0005 xx190)P=-0.0005P^(2)]:}Now By using Bernouli sequation.{:[v=y^(1-n)=>v=p^(1-2.quad" (Highest power of "p=2" ) ")],[:.quad v=p^(-1)=(1)/(p.)]:}Diflerentaating w.r.t on both sides.{:[(dv)/(dt)=-1*P^(-1-1)(dP)/(dt)quad(:'(d)/(dx)(xn)=n+x^(n-1)).],[(dv)/(dt)=-(1)/(p^(2))(dp)/(dt)],[" noo "{:(1)/((1)/(p^(2))dt))=-(dv)/(ddt-1)]:}7rom (1) (1)/(p^(2))*(dP)/(dt)-((0.0005 xx190)P)/(P^(2))=(-0.0005P^(2))/(P^(2)):.(1)/(p^(2))(dp)/(dt)-(0.0005 xx190)((1)/(p))=-0.0005.substulue@{:[-(dv)/(dt)-(0.0005 xx190)v=-0.0005(:'V=p^(-1)=(1)/(p).:}],[:.(dv)/(dt)+(0.0005 xx190)v=0.0005],[xi(1)/(p^(2))(dp)/(dt)=-(dV)/(dt)" from(B). "]:}It is in the form of{:[(dy)/(dx)+Py=Q.],[:.quad I" tegration factor "=e^(int Pdx.)]:}& Solution will be yIF=int Q=Fdx;Now (dV)/(dt)+(0.0005 xx190)V=0.0005=>{:[P=0.0005 xx190],[Q=0.0005]:}:. Integration Factor =e^(ȷpdx){:[" I. F "=e^(j 0.0005 xx190 t).],[=e^((0.0005 xx190)]//t)],[" I.F "=e^((0.0005 xx190)t).]:}The solution will be like this{:[" V. I.F " ... See the full answer