Question

Solve the seperable differential equation for.
\frac(dy)(dx) = \frac(1+x)(xy^(6)) ; \ \ x \gt 0
Use the following initial condition: y(1) = 5 .

Answer 1

\[\Large \frac{dy}{dx}=\frac{1+x}{xy^6}, \qquad\qquad x\gt0, \qquad\qquad y(1)=5\]

Answer 2

What part are you stuck on maria? :o
Understand the separation portion of it?\[\Large y^6\;dy=\frac{1+x}{x}\;dx\]

Answer 3

No I don't!!! I forgot ittt

Answer 4

\[\Large \frac{dy}{dx}=\frac{1+x}{xy^6}\]So we want all of our `y and dy` on one side of the equation with all of our `x and dx` on the other side :o

So we start by multiplying both sides by y^6.\[\Large y^6\frac{dy}{dx}=\frac{1+x}{x}\]Right? +_+

Answer 5

so what do I have to plug in to get y=? because that part makes sense I just don't know how to use that information further

Answer 6

You need to integrate which will get rid of the differentials.
Have you done that part yet? :o\[\Large \int\limits y^6\;dy=\int\limits \frac{1+x}{x}\;dx\]

Answer 7

1/7y^7=(ln|x|)+x

Answer 8

\[\Large \frac{1}{7}y^7=\ln|x|+x+C\]
Ok good :) Now we can use our initial condition solve for our unknown constant.

Answer 9

\[\Large y(1)=5\]Plugging in gives us,\[\Large \frac{1}{7}5^7=\ln|1|+1+C\]

Answer 10

11160.71=10.320155+c

Answer 11

Nooo, no decimals! :O
And what happened on the right side there..?
Recall: ln1=0

Answer 12

111.60.71=1+C

Answer 13

11160=C

Answer 14

fine fine fine :) have your sloppy decimals lol
that looks good.

So now we plug our c back in. and since we're able to, we should probably solve for y.

Answer 15

\[\Large \frac{1}{7}y^7=\ln|x|+x+11160\]

Answer 16

\[\Large y=?\]