## anonymous one year ago Write and then solve for y = f(x) the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y^4."

1. Michele_Laino

hint: two quantities, say A and B are inversely proportional if the subsequent condition olds: A*B = k where k is a constant

2. Michele_Laino

holds*

3. anonymous

dy/dx = ky^4 ?

4. Michele_Laino

no, since that formula expresses the direct proportionality

5. Michele_Laino

hint: we can write this: A= dy/dx, and B=y^4

6. anonymous

dy/dx = k/y^4

7. Michele_Laino

correct!

8. anonymous

okay and now int(dy y^4) = int(dx k) (y^5)/5 = (x^2)/2 + C

9. anonymous

$y = \sqrt[5]{\frac{ 5x^2 }{ 2 } + C}$

10. Michele_Laino

I got this: $\Large \begin{gathered} \int {{y^4}dy} = \int {kdx} \hfill \\ \hfill \\ \frac{{{y^5}}}{5} = kx + C \hfill \\ \end{gathered}$

11. anonymous

doesn't the right side turn into $\frac{ kx^2 }{ 2 } + C$

12. anonymous

no wait i see what i did wrong

13. anonymous

okay so then $y = \sqrt[5]{5kx + C}$

14. Michele_Laino

correct!

15. Michele_Laino

sorry! I had to specify this: $\Large C \in \mathbb{R}$ since you are searching for a real funtion

16. anonymous

thank you so much :D

17. Michele_Laino

:)

18. anonymous

quick question what does the $\epsilon$ mean

19. Michele_Laino

in general, in mathematical analysis, the mathematicians use \epsilon in order to indicate a small and positive quantity, which, in some case goes to zero

20. anonymous

okay , double thank you :D

21. Michele_Laino

:)