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Integral Calc -Need Help
Topic Started: Jan 5 2011, 05:42 PM (333 Views)
surfer
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I understand regular integrals with polynomials after learning about it today, but I've sort of forgotton and I've got this problem that involves more than basic integrals and derivatives.

Question:

You are given that d2y/dx^2 = 3sin(x) - 4 cos(x) and that y=7 and y'=2 when x=0 for both. I need to find the differential equation from this.
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Nova
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This is a fairly complex problem but I've seen worse. Here's an answer for you.

Take the integral of the second derivative. ⌠(3sinx - 4 cosx)dx is y'= -3cosx - 4sinx + c (remember to include +c because I tend to forget)

Take y' and x=0 and sub them in and you find that c = 5.
Do your first derivative equation is y' = -3cosx - 4sinx - 5, but you are not done yet.

Take the integral of the first derivative that you found. ⌠(-3cosx - 4sinx - 5)dx and get y = [-3sinx + 4cosx + 5x + c].
Set it equal to 7 and sub 0 in for c. c = 3

From this you can tell that the differential equation is y = -3sinx + 4cosx + 5x + 3.

Hope this helps!
Edited by Nova, Jan 5 2011, 06:01 PM.
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surfer
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This makes sense. I did this and tryed everything you did on a new problem and got a nice answer. Thanks a lot!
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surfer
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Is it possible to take the Integral of a function that uses F(-1) = 7 as it's info?
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Nova
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Polar98 CO
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No. -1 happens to be the only number that doesn't work in integral calculus. Here's why:

When finding the integral of a derivative take for example ⌠(2x - [1/x])dx. You add the constant right away then add a power to x making it 2x. Then you have to divide by the power which is 2. So you get x^2. Now you look at the second part where it is -1. Take th constant out, raise the power and you get -x^0 = -1 then you need to divide by -1+1 which is zero. This make the function undefined and gives you no result.

It is possible to integrate this with natural logarithms, but you've probably not studied them yet. For your purposes, this is impossible.
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surfer
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OK. I see how that works. Just wondering.
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Swift
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I SEE NOTHING BUT LITTLE SQUIGGLYS AND STUFF I DONT UNDERSTAND AHHHHHHHHHHHHHHHHHHHH!!!!!
Never a failure, Always a lesson.

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surfer
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We are taking the slope of the slope of the tangent line (the second derivative) and solving for the exact original function. This one happens to involve triganometry.

Nova: This may be a stupid question but what is the integral of the derivative of (x^3-8)^1/3?
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Nova
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Polar98 CO
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⌠(d/dx)(x^3 - 8)^1/3dx is actually just y = (x^3 - 8)^1/3dx + c.

Taking the integral of the derivative is just like multiplying 2/1 by 1/2. You get what you started with accept for integrals you get +c. If it is the derivative of the integral, there is no +c.
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surfer
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Oh OK, because I had a problem like that and thought it was a foolish question, but yes. Thanks for the reponse so quickly.
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