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."

Question

michele_laino · Answer

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

michele_laino · Answer

holds*

anonymous · Answer

dy/dx = ky^4  ?

michele_laino · Answer

no, since that formula expresses the direct proportionality

michele_laino · Answer

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

anonymous · Answer

dy/dx = k/y^4

michele_laino · Answer

correct!

anonymous · Answer

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

anonymous · Answer

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

michele_laino · Answer

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

anonymous · Answer

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

anonymous · Answer

no wait i see what i did wrong

anonymous · Answer

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

michele_laino · Answer

correct!

michele_laino · Answer

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

anonymous · Answer

thank you so much :D

michele_laino · Answer

:)

anonymous · Answer

quick question what does the $$\epsilon $$ mean

michele_laino · Answer

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

anonymous · Answer

okay , double thank you :D

michele_laino · Answer

:)