anonymous
  • anonymous
If dy/dx = sin^2( piy/4) and y = 1 when x = 0, then find the value of x when y = 3.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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freckles
  • freckles
do you know how to solve the given differential equation?
anonymous
  • anonymous
im not too sure how to manipulate dy/dx
anonymous
  • anonymous
i know i have to isolate the variables and integrate and solve for y right?

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freckles
  • 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
  • anonymous
well the right side is just x but im not to sure on integrating the left side
freckles
  • freckles
do you know the derivative of cot(x)?
anonymous
  • anonymous
-csc^2(x)?
freckles
  • freckles
\[\frac{d}{dy} \cot(u)=-\csc^2(u) \cdot \frac{du}{dy} \\ \text{ where } u=u(y)\]
freckles
  • 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
  • freckles
do you see how to continue
anonymous
  • anonymous
why did you inverse the pi/4
freckles
  • freckles
because if du=pi/4 dy then dy=4/pi du
freckles
  • freckles
i just multiplied 4/pi on both sides to isolate dy
anonymous
  • anonymous
okay okay hold up slow down , why are we integrating csc^2(piy/4)
freckles
  • freckles
because that was the left hand side of your equation once you separated the variables
anonymous
  • anonymous
okay okay i see, give me a second to get my thoughts together
freckles
  • freckles
\[\frac{1}{\sin(x)}=\csc(x)\]
anonymous
  • anonymous
yea but the sine was squared wouldn't that affect that
freckles
  • 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
  • freckles
\[(\frac{1}{\sin(x)})^2=\csc^2(x) \\ \frac{1}{\sin^2(x)}=\csc^2(x)\]
anonymous
  • anonymous
okay sorry i was thinking the square would be negative because it was in the denominator
anonymous
  • anonymous
i see what you mean though
anonymous
  • anonymous
okay so we are integrating \[\csc^2(\frac{ \pi y }{ 4 })\]
anonymous
  • anonymous
oh you were doing substitution , i see
anonymous
  • 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
  • freckles
on what? that is just -4/pi*cot(u)+C where u=pi/4y
anonymous
  • anonymous
\[\frac{ 4 }{ \pi } \int\limits (-\ln(\csc(x) + \cot(x)) + C)^2\]
anonymous
  • anonymous
is that right?
anonymous
  • anonymous
minus the integral sign
freckles
  • freckles
where did you get that answer
freckles
  • freckles
remember you yourself said the d/dx cot(x) =-csc^2(x)
freckles
  • freckles
or you could say -d/dx cot(x)=csc^2(x)
anonymous
  • anonymous
anonymous
  • anonymous
are these wrong then?
freckles
  • 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
  • freckles
nope those all look great
anonymous
  • anonymous
arghh i feel stupid
freckles
  • freckles
anyways we also have to integrate the other side
freckles
  • freckles
\[\frac{-4}{\pi} \cot(\frac{\pi}{4} y)+C=x\]
anonymous
  • anonymous
-_- i was just typing that, okay so now just isolate y right
freckles
  • freckles
your second job is to find C for the point (x=0,y=1)
freckles
  • freckles
i would not solve for y i think that would cause more work then needed
freckles
  • freckles
besides you want to find x
freckles
  • freckles
for when y=3
freckles
  • freckles
which is the third part of the problem third and final
anonymous
  • anonymous
\[C = \pi\]
anonymous
  • anonymous
- pi
freckles
  • 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
  • anonymous
:(
freckles
  • freckles
it's okay... no frowns here!
anonymous
  • anonymous
alright well c = 4
freckles
  • 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
  • freckles
well no
freckles
  • freckles
c=4/pi
anonymous
  • anonymous
okay let me just take a breath , my frustration is getting the better of me here
anonymous
  • anonymous
okay so now we just plug c into the solution and plug y = 3 and find x right
freckles
  • freckles
right
freckles
  • freckles
i will let you do that and I will come back in check in like 5 minutes or less
anonymous
  • anonymous
okay im getting 8/pi
freckles
  • freckles
that sounds right
freckles
  • freckles
4/pi+4/pi 2*4/pi 8/pi
anonymous
  • anonymous
okay great , thanks for the help sorry for being such a dummy :P
freckles
  • 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
  • freckles
you aren't a dummy probably just frustrated and trying to learn a new subject
anonymous
  • anonymous
no im goo thanks again !
freckles
  • freckles
the one part is hard enough being frustrated makes it tad harder
freckles
  • freckles
anyways peace and i gave you a medal to award you for your effort
anonymous
  • anonymous
:) night

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