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

In summary, the task is to solve the initial value problem: f'(x)=(x^2-1)/x^5 with the given initial condition f(1/2)=3. The integration process is correct, but the constant term needs to be added. After finding the value of C, it can be substituted to solve for the original function.
  • #1
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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  • #2
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 [tex]\frac{-1}{2x^2}+\frac{1}{4x^4}[/tex]. 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.
 
  • #3
Ohhh so I'm solving for C. Thanks!
 

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

What is an initial value problem?

An initial value problem is a type of differential equation that involves finding a function or equation that satisfies certain conditions at a specific point, known as the initial value. In this case, the initial value is f(1/2)=3.

How do you solve an initial value problem using integration?

To solve an initial value problem using integration, we first need to rewrite the given equation using the derivative notation, f'(x). Then, we integrate both sides of the equation and solve for the constant of integration using the given initial value.

What is the general process for solving initial value problems?

The general process for solving initial value problems involves rewriting the equation using the derivative notation, integrating both sides, and solving for the constant of integration using the given initial value. This process may vary depending on the type of differential equation and initial value given.

How do you determine the constant of integration in initial value problems?

The constant of integration is determined by using the given initial value. This value is substituted into the equation after integration, and then solved for the constant. In this case, the initial value of f(1/2)=3 is used to find the constant of integration.

Can initial value problems be solved using other methods besides integration?

Yes, initial value problems can also be solved using other methods such as separation of variables, substitution, and numerical methods. However, integration is a common and widely used method for solving initial value problems.

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