Question

How do you solve Bernoulli differential equation y'-5y=3(e^t)y^2 with the conditions of y(0)=5? Please help.

Answer 1

Use the substitution \(w=y^{1-n}\) to convert it into a differential euqation in terms of "w"

Answer 2

However, you need to get it into standard form first! So divide everything by \(y^2\)!
Essentially, you're going to massage the differential equation into a linear first oder differential equation.

Answer 3

So I get y^-2(y')-5y^-1=3e^5t

Answer 4

And then I have to substitute w? I'm still a bit confused...

Answer 5

You have: \(\frac{1}{y^2} y'-\frac{5}{y}=e^t\)
now use substitution, \(w=y^{-1}\) and find \(w'\) you should now see that you get a nice linear differential.

Answer 6

Yeah, so I get -w'-5w=e^t. Wait, what happened to the 3 and the 5th power e?

Answer 7

Oops, forgot about it, but it's still there. Lol. It's just a constant.

Answer 8

Oh ok. So once I get w an the derivative of it and substitute it into the function. Then do I find the integrating factor?

Answer 9

Yep! yOU SHOuld know what to do from here. I think you learned how to solve linear differential eequation from day 1.

Answer 10

umm just a quick question. The integrating factor for this turns out to be e^(-5t) right?

Answer 11

\(u(x) = e^{\int 5dt}\)

Answer 12

I ended up getting 1/((3/10)-(1/10)e^(-10t)). Does that look correct?

Answer 13

i can't even read that. Lol

Answer 14

It looks like this.

Answer 15

Is that your final solution?

Answer 16

That doesn't look right to me because you're missing the constant in your solution...

Answer 17

Oh I solved for the constant by using the given y(0)=5

Answer 18

And the answer I gave is what it looks like after having solved for C.

Answer 19

No, still wrong. Lol. If your initial condition is y(0)=5....also your exponential term is wrong. How did you get -10?

Answer 20

I multiplied the integrating factor of e^(5t) to the equation with the substituted w. I think I may have done it incorrectly.

Answer 21

\( \large\int [e^{5t}w]' = \int -3e^{6t}dt\)

Answer 22

I got this.

Answer 23

My math could be off or could be yours. But either way, close enough I guess. Lol.