Question

differential equation problem

Answer 1

the population P of a flock of birds is growing exponentially so that dP/dx=20e^.05x
where x is time in years. Find P in terms of x if there were 20 birds in the flock initially.

Answer 2

\[\frac{ dP }{ dx }= 20e ^{.05x}\]

Answer 3

This is pretty easy, just integrate with respect to x.

Recall \[\int\limits e^{ax+b} dx= \frac{ 1 }{ a } e^{ax+b}\]

Answer 4

with the constant of integration

Answer 5

1/20 e^20x?

Answer 6

\[20 \int\limits e^{0.5x}dx = \int\limits dP\]

Answer 7

\[\frac{ dP }{ dx } = 20e^{0.05x}; P = \int\limits 20e^{0.05x} dx = 20\int\limits e^{0.05x }dx \]

\[P = 20\int\limits e^{0.05x} dx = \frac{ 20 }{ 0.05 } e^{0.05x} +C\]

Answer 8

see we have divided by the coefficient of the power (0.05) but the exponential is retained

Answer 9

i got it. but then in the next problem it asks find the particular solution if there were 100 birds in the flock after 2 years

Answer 10

i dont understand how to do that

Answer 11

using the first problem

Answer 12

sub in x = how many years you need into your answer

Answer 13

but of course you need to find C first. It says P = 20 initially (i.e. when x = 0)
so sub in P = 20 and x = 0 to find C, then replace the C in the equation

Answer 14

the second one sub in P = 100 and x = 20 to find the C for that question

Answer 15

c= -380 for the first

Answer 16

you mean sub in x = 2 for the second one right?

Answer 17

\[400e ^{0.05(2)}+c=100\]

Answer 18

sorry.05

Answer 19

you would plug in that into your original equation you just found to find out what c is for your initial condition

Answer 20

got it thank you