separation of variables to find k and y(t)

Question

I need help solving this problem:

 

A function y(t) is a solution of
dy/dt+t^(k)*y=0.

Suppose that y(0)=1 and y(1)=e−5
Find k and find y(t).

 

can anyone please help?

Answer 1

Noah,

 

Since you have already identified the equation as separable, you should be able to easily solve it, assuming k is a constant.

 

Separating the variables,

 

dy/y = –t^k dt .

 

Now integrate both sides:

 

ln |y| = –t^k+1/(k+1) + C.

 

Plug in the first boundary conditions:

 

y(0) = 1 implies 0 = 0 + C, so C = 0.

 

Now, you have the particular solution

 

ln |y| = –t^k+1/(k+1).

 

Plug in the second boundary condition to solve for k:

 

ln |e-5| = -1/(k+1)

 

k+1 = –1/ln (5-e)   (since e-5 < 0, its absolute value is its opposite)

 

k = –1 – 1/ln(5-e).

 

Next, plug this into the solution and solve for y:

 

ln |y| = –t^{-1/ln (5-e)}/(–1/ln (5-e)) = t^{–1/ln (5-e)} ln (5-e) = ln (5-e)/t^{1/ln (5-e)} .

 

|y| = exp {ln (5-e)/t^{1/ln (5-e)}}

 

y = exp {ln (5-e)/t^{1/ln (5-e)}}.

 

(Remember that exp(x) = e^x.) There are other ways to write this more nicely. For example, you could use algebra and properties of exponentials to put the answer in the form

 

y = (5-e)^[t^{-1/ln (5-e)}].

 

This is hard to show clearly in WyzAnt, but it is a little easier to calculate with. (For some reason my font size shrunk, so I hope you are still able to read this.)

 

Let me know if you have any questions about my solution.