If dy/dx = sin^2( piy/4) and y = 1 when x = 0, then find the value of x when y = 3.

- anonymous

If dy/dx = sin^2( piy/4) and y = 1 when x = 0, then find the value of x when y = 3.

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- freckles

do you know how to solve the given differential equation?

- anonymous

im not too sure how to manipulate dy/dx

- anonymous

i know i have to isolate the variables and integrate and solve for y right?

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## More answers

- freckles

\[\frac{dy}{dx}=\sin^2(\frac{\pi}{4} y) \\ \frac{dy}{\sin^2(\frac{\pi}{4} y)}=dx \\ \]
like do you know how to ingrate both sides of this equation?

- anonymous

well the right side is just x but im not to sure on integrating the left side

- freckles

do you know the derivative of cot(x)?

- anonymous

-csc^2(x)?

- freckles

\[\frac{d}{dy} \cot(u)=-\csc^2(u) \cdot \frac{du}{dy} \\ \text{ where } u=u(y)\]

- freckles

\[\int\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits \csc^2(u) \frac{4}{\pi} du\]

- freckles

do you see how to continue

- anonymous

why did you inverse the pi/4

- freckles

because if du=pi/4 dy
then dy=4/pi du

- freckles

i just multiplied 4/pi on both sides to isolate dy

- anonymous

okay okay hold up slow down , why are we integrating csc^2(piy/4)

- freckles

because that was the left hand side of your equation once you separated the variables

- anonymous

okay okay i see, give me a second to get my thoughts together

- freckles

\[\frac{1}{\sin(x)}=\csc(x)\]

- anonymous

yea but the sine was squared wouldn't that affect that

- freckles

so you know that equation I just mentioned holds
don't you think the equation will still hold if you square both sides

- freckles

\[(\frac{1}{\sin(x)})^2=\csc^2(x) \\ \frac{1}{\sin^2(x)}=\csc^2(x)\]

- anonymous

okay sorry i was thinking the square would be negative because it was in the denominator

- anonymous

i see what you mean though

- anonymous

okay so we are integrating \[\csc^2(\frac{ \pi y }{ 4 })\]

- anonymous

oh you were doing substitution , i see

- anonymous

okay so back to \[\int\limits\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits\limits \csc^2(u) \frac{4}{\pi} du \]
should be do another substitution?

- freckles

on what? that is just -4/pi*cot(u)+C
where u=pi/4y

- anonymous

\[\frac{ 4 }{ \pi } \int\limits (-\ln(\csc(x) + \cot(x)) + C)^2\]

- anonymous

is that right?

- anonymous

minus the integral sign

- freckles

where did you get that answer

- freckles

remember you yourself said the d/dx cot(x) =-csc^2(x)

- freckles

or you could say -d/dx cot(x)=csc^2(x)

- anonymous

##### 1 Attachment

- anonymous

are these wrong then?

- freckles

\[\int\limits\limits\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits\limits\limits \csc^2(u) \frac{4}{\pi} du \\ \frac{-4}{\pi} \cot(u)+C \\ \frac{-4}{\pi} \cot(\frac{\pi}{4}y)+C\]

- freckles

nope those all look great

- anonymous

arghh i feel stupid

- freckles

anyways we also have to integrate the other side

- freckles

\[\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x\]

- anonymous

-_- i was just typing that, okay so now just isolate y right

- freckles

your second job is to find C for the point (x=0,y=1)

- freckles

i would not solve for y
i think that would cause more work then needed

- freckles

besides you want to find x

- freckles

for when y=3

- freckles

which is the third part of the problem
third and final

- anonymous

\[C = \pi\]

- anonymous

- pi

- freckles

\[\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x \\ (0,1)=(x,y) \\ \frac{-4}{\pi}\cot(\frac{\pi}{4}(1))+C=0 \\ \frac{-4}{\pi} \cot(\frac{\pi}{4})+C=0 \\ \frac{-4}{\pi}(1)+C=0 \\ \frac{-4}{\pi}+C=0\]

- anonymous

:(

- freckles

it's okay... no frowns here!

- anonymous

alright well c = 4

- freckles

\[\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x \\ \frac{-4 }{\pi} \cot(\frac{\pi}{4} y)+\frac{4}{\pi}=x\]
last step find x for when y is 3

- freckles

well no

- freckles

c=4/pi

- anonymous

okay let me just take a breath , my frustration is getting the better of me here

- anonymous

okay so now we just plug c into the solution and plug y = 3 and find x right

- freckles

right

- freckles

i will let you do that
and I will come back in check in like 5 minutes or less

- anonymous

okay im getting 8/pi

- freckles

that sounds right

- freckles

4/pi+4/pi
2*4/pi
8/pi

- anonymous

okay great , thanks for the help
sorry for being such a dummy :P

- freckles

i'm going to go for tonight after this problem
sleep time
but before i go do you have any last questions on this problem?

- freckles

you aren't a dummy
probably just frustrated and trying to learn a new subject

- anonymous

no im goo thanks again !

- freckles

the one part is hard enough
being frustrated makes it tad harder

- freckles

anyways peace
and i gave you a medal to award you for your effort

- anonymous

:) night

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