Question

I need help with the substitution part of the differential equation (ln(y))^7*dy/dx=(x^7)y

Answer 1

So, you already got to this point:
\( \displaystyle \frac{( \ln y )^7}{y} \; \text{d}y = x^7 \; \text{d}x \)
?

Answer 2

Yes and I am not sure if I could integrate both sides or if I have to substitute anything in.

Answer 3

Well, the right side integrates fairly nicely.

The left side, if we integrate this:
\( \displaystyle \int \frac{ (\ln y)^7 }{y} \; \text{d}y \)
Recall that the derivative of ln y = 1/y. So, we have the function and its derivative here in the integral...

Answer 4

But since it is raised to the power of 7 make it so you can do that?

Answer 5

Even tho it is raised to that power, we are able to make this substitution as long as we are consistent with our substitution, u = ln y; du = 1/y dy. It becomes:
\( \displaystyle \int u^7 \; \text{d}u \)
Because we have everything covered in the problem, essentially.

Answer 6

The du is dy/y, and u^7 is the same as (ln y)^7

Answer 7

Ok I think was making a mistake with the substitution thanks.

Answer 8

You're welcome! :)