Question

solve the initial value problem: (1/x)dy/dx=(ysinx)/(y^2+1), y(pi)=1

I separate both sides, then I integrate which leaves me with, 1/2y^2+lny=-cosx+c but I am lost after this. I've tried reading the book, but it was no help. Please I really need help.

Answer 1

hmm how did you get the other side (the right hand side)

Answer 2

you did integrate x *sin(x)?

Answer 3

by separation of variables you did get:
\[\frac{y^2+1}{y} dy =x sin(x) dx \]right ?

Answer 4

no I did not get xsin(x)dx, I got sin(x)dx

Answer 5

well don't you have 1/x on that one side?

Answer 6

\[\frac{1}{x} \frac{dy}{dx}=\frac{y \sin(x)}{y^2+1} \]
multiply x dx on both sides giving you
\[dy=\frac{y \sin(x)}{y^2+1} x dx\]
multiply both sides by (y^2+1)/y

Answer 7

Ah , I suppose that is where I messed up at.

Answer 8

Let me know when you have fixed that part
and I will help you with the rest or this part if you need further help on this particular part

Answer 9

thank you

Answer 10

So now I get 1/2y^2+lny=-xcosx+sinx+c
I know that I am supposed to solve for y so that I can set the equation equal to the initial condition of y(pi)=1, but I do not know how to solve for y in this problem.

Answer 11

a lot better
I just want to make one change
\[\frac{y^2}{2}+\ln|y|=-xcos(x)+\sin(x)+C\]


you don't have to solve for y

now replace x with pi and y with 1 because y(pi)=1 says so

Answer 12

oh, ok so with that then I get c=1/2+1

Answer 13

hmmm...doesn't seem right

Answer 14

so you replaced x with pi and y with 1
so you have
\[\frac{1}{2}+0=-\pi \cos(\pi)+C\]

Answer 15

and of course cos(pi) is -1

Answer 16

but don't forget that is being multiplied by -1

Answer 17

ok I see I forgot about the pi in front of -cosx again.
now I get c=1/2-pi

Answer 18

agreed that is good stuff
replace that c above with 1/2-pi
and you are done

Answer 19

oh man so many mistakes I made. Thank you.

Answer 20

i only solve for y when it is feasible

Answer 21

i think you did okay

Answer 22

plus I didn't know I could plug in the 1 for y that was a big help

Answer 23

yep y(A)=B
mean replace x (or t) with A
and replace y with B

Answer 24

I said or t because t is commonly used too