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Beez

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Hi, I found the same problem I needed a help to solve somewhere in this forum. However, I could not reach the answers only with the help provided on that page. I would really appreciate it if someone could offer me some help.

P: Let c be a positive number. A differential equation of the form: dy/dt = ky^(1+c)

where k is a positive constant, is called a doomsday equation because the equation in the expression ky^(1+c) is larger than that for natural growth (that is, ky).

(a) Determine the solution that satisfies the initial condition y(0)=y(subzero)

What I did was

dy/y^-(1+c)=k dt Integrate both sides I got

y(t)=1/[ck(T-t)]^(1/c) for some constant T

Is this correct?

(ba) Show that there is a finite time t = ta (doomsday) such that lim(t->T-) wy(t) = infinity

For the equation, infinity = (limt->T-)1/[ck(T-t)]^1/c, when t approaches to T, T=t or T-t=0, which makes the denominator 0, hence the value of the equation becomes infinity.

Is this what I need to say, or should I get the exact value of t (can I?)?

(c) An especially prolific breed of rabbits has the growth term ky^(1.01). If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

Since y^(1.01), c=0.1. and Y(3)=16, By substituting those numbers to the equation to obtain the value of k.

16^0.1=1/[0.1*k(3)]^(1/0.1)

[0.3k]^10=1/1.32

0.3k=(1/1.32)^(1/10)

k=0.9726

This time use wy(0)=2 to get the value of T

2=1/[0.1*0.9726*T]^(1/10)

[0.1*0.9726*T]^(1/10)=1/2

[0.1*0.9726*T]=(1/2)^(1/10)

T=0.9330/0.09726=(approx)9.6months

How does that sound?

I have no confident with these solutions, especially (c).

Someone, please help me!

P: Let c be a positive number. A differential equation of the form: dy/dt = ky^(1+c)

where k is a positive constant, is called a doomsday equation because the equation in the expression ky^(1+c) is larger than that for natural growth (that is, ky).

(a) Determine the solution that satisfies the initial condition y(0)=y(subzero)

What I did was

dy/y^-(1+c)=k dt Integrate both sides I got

y(t)=1/[ck(T-t)]^(1/c) for some constant T

Is this correct?

(ba) Show that there is a finite time t = ta (doomsday) such that lim(t->T-) wy(t) = infinity

For the equation, infinity = (limt->T-)1/[ck(T-t)]^1/c, when t approaches to T, T=t or T-t=0, which makes the denominator 0, hence the value of the equation becomes infinity.

Is this what I need to say, or should I get the exact value of t (can I?)?

(c) An especially prolific breed of rabbits has the growth term ky^(1.01). If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

Since y^(1.01), c=0.1. and Y(3)=16, By substituting those numbers to the equation to obtain the value of k.

16^0.1=1/[0.1*k(3)]^(1/0.1)

[0.3k]^10=1/1.32

0.3k=(1/1.32)^(1/10)

k=0.9726

This time use wy(0)=2 to get the value of T

2=1/[0.1*0.9726*T]^(1/10)

[0.1*0.9726*T]^(1/10)=1/2

[0.1*0.9726*T]=(1/2)^(1/10)

T=0.9330/0.09726=(approx)9.6months

How does that sound?

I have no confident with these solutions, especially (c).

Someone, please help me!

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