Solving Initial Value Problems Using Integration | f'(x)=(x^2-1)/x^5, f(1/2)=3

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To solve the initial value problem f'(x)=(x^2-1)/x^5 with f(1/2)=3, integration is required to find the original function. The integration yields -1/(2x^2) + 1/(4x^4) plus a constant C. It’s crucial to include the constant of integration when solving, which can be determined by substituting the given values into the equation. The confusion arose from not recognizing the need to solve for C after integrating. Properly applying these steps will lead to the correct solution for the initial value problem.
howsockgothap
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Homework Statement



Solve each of the following initial value problems: a) f'(x)=(x2-1)/x5 f(1/2)=3

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The Attempt at a Solution


I guess my problem with this is I'm not 100% sure what I'm being asked to do. I know I need to use integration to find the original function. Ok no prob I got -1/2x-2+1/4-4. So after that am I just supposed to plug in 3=the original function, substituting 1/2 for x? That doesn't seem right since I'm not getting 3... Am I doing the question incorrectly or have I done my integration wrong?
 
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howsockgothap said:
Solve each of the following initial value problems: a) f'(x)=(x2-1)/x5

You're leaving out the whole problem. Did the question say f(1/2) = 3? Or f(3) = 1/2? As far as your integral goes, it looks fine to me if you mean \frac{-1}{2x^2}+\frac{1}{4x^4}. However, don't forget to add a constant everytime you take an integral, so just put a +C at the end. After that, just plug in the x and f(x) values to solve for C.
 
Ohhh so I'm solving for C. Thanks!
 

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