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anonymous
 one year ago
If dy/dx = sin^2( piy/4) and y = 1 when x = 0, then find the value of x when y = 3.
anonymous
 one year ago
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
 one year ago
Best ResponseYou've already chosen the best response.4do you know how to solve the given differential equation?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0im not too sure how to manipulate dy/dx

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i know i have to isolate the variables and integrate and solve for y right?

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\frac{dy}{dx}=\sin^2(\frac{\pi}{4} y) \\ \frac{dy}{\sin^2(\frac{\pi}{4} y)}=dx \\ \] like do you know how to ingrate both sides of this equation?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well the right side is just x but im not to sure on integrating the left side

freckles
 one year ago
Best ResponseYou've already chosen the best response.4do you know the derivative of cot(x)?

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\frac{d}{dy} \cot(u)=\csc^2(u) \cdot \frac{du}{dy} \\ \text{ where } u=u(y)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\int\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits \csc^2(u) \frac{4}{\pi} du\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.4do you see how to continue

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0why did you inverse the pi/4

freckles
 one year ago
Best ResponseYou've already chosen the best response.4because if du=pi/4 dy then dy=4/pi du

freckles
 one year ago
Best ResponseYou've already chosen the best response.4i just multiplied 4/pi on both sides to isolate dy

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay okay hold up slow down , why are we integrating csc^2(piy/4)

freckles
 one year ago
Best ResponseYou've already chosen the best response.4because that was the left hand side of your equation once you separated the variables

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay okay i see, give me a second to get my thoughts together

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\frac{1}{\sin(x)}=\csc(x)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yea but the sine was squared wouldn't that affect that

freckles
 one year ago
Best ResponseYou've already chosen the best response.4so you know that equation I just mentioned holds don't you think the equation will still hold if you square both sides

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[(\frac{1}{\sin(x)})^2=\csc^2(x) \\ \frac{1}{\sin^2(x)}=\csc^2(x)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay sorry i was thinking the square would be negative because it was in the denominator

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i see what you mean though

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so we are integrating \[\csc^2(\frac{ \pi y }{ 4 })\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh you were doing substitution , i see

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay 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
 one year ago
Best ResponseYou've already chosen the best response.4on what? that is just 4/pi*cot(u)+C where u=pi/4y

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\frac{ 4 }{ \pi } \int\limits (\ln(\csc(x) + \cot(x)) + C)^2\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0minus the integral sign

freckles
 one year ago
Best ResponseYou've already chosen the best response.4where did you get that answer

freckles
 one year ago
Best ResponseYou've already chosen the best response.4remember you yourself said the d/dx cot(x) =csc^2(x)

freckles
 one year ago
Best ResponseYou've already chosen the best response.4or you could say d/dx cot(x)=csc^2(x)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0are these wrong then?

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\int\limits\limits\limits_{}^{} \csc^2(\frac{\pi}{4} y) dy \\ u=\frac{\pi}{4} y \\ du=\frac{\pi}{4} dy \\ \int\limits\limits\limits \csc^2(u) \frac{4}{\pi} du \\ \frac{4}{\pi} \cot(u)+C \\ \frac{4}{\pi} \cot(\frac{\pi}{4}y)+C\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.4nope those all look great

freckles
 one year ago
Best ResponseYou've already chosen the best response.4anyways we also have to integrate the other side

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\frac{4}{\pi} \cot(\frac{\pi}{4} y)+C=x\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0_ i was just typing that, okay so now just isolate y right

freckles
 one year ago
Best ResponseYou've already chosen the best response.4your second job is to find C for the point (x=0,y=1)

freckles
 one year ago
Best ResponseYou've already chosen the best response.4i would not solve for y i think that would cause more work then needed

freckles
 one year ago
Best ResponseYou've already chosen the best response.4besides you want to find x

freckles
 one year ago
Best ResponseYou've already chosen the best response.4which is the third part of the problem third and final

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\frac{4}{\pi} \cot(\frac{\pi}{4} y)+C=x \\ (0,1)=(x,y) \\ \frac{4}{\pi}\cot(\frac{\pi}{4}(1))+C=0 \\ \frac{4}{\pi} \cot(\frac{\pi}{4})+C=0 \\ \frac{4}{\pi}(1)+C=0 \\ \frac{4}{\pi}+C=0\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.4it's okay... no frowns here!

freckles
 one year ago
Best ResponseYou've already chosen the best response.4\[\frac{4}{\pi} \cot(\frac{\pi}{4} y)+C=x \\ \frac{4 }{\pi} \cot(\frac{\pi}{4} y)+\frac{4}{\pi}=x\] last step find x for when y is 3

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay let me just take a breath , my frustration is getting the better of me here

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so now we just plug c into the solution and plug y = 3 and find x right

freckles
 one year ago
Best ResponseYou've already chosen the best response.4i will let you do that and I will come back in check in like 5 minutes or less

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay im getting 8/pi

freckles
 one year ago
Best ResponseYou've already chosen the best response.44/pi+4/pi 2*4/pi 8/pi

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay great , thanks for the help sorry for being such a dummy :P

freckles
 one year ago
Best ResponseYou've already chosen the best response.4i'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
 one year ago
Best ResponseYou've already chosen the best response.4you aren't a dummy probably just frustrated and trying to learn a new subject

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no im goo thanks again !

freckles
 one year ago
Best ResponseYou've already chosen the best response.4the one part is hard enough being frustrated makes it tad harder

freckles
 one year ago
Best ResponseYou've already chosen the best response.4anyways peace and i gave you a medal to award you for your effort
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