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

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SUMMARY

The discussion focuses on solving the initial value problem defined by the differential equation f'(x) = (x² - 1)/x⁵ with the condition f(1/2) = 3. The correct approach involves integrating the function to obtain f(x) = -1/(2x²) + 1/(4x⁴) + C, where C is a constant determined by substituting the initial condition. The participant initially misunderstood the integration process and the role of the constant C, which is crucial for finding the specific solution that satisfies the initial condition.

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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

Homework Equations





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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