Solving for x: Questioning a Puzzling Step

[zorro]

Homework Statement

These are the steps in my book for a question-

f(x) x f¹(x) = 1

∫f(x) x d{f(x)} = ∫1dx

{f(x)}²/2 + C = x

I don't get how is second step possible.

Say we have a function f(x) = x²

By the above steps,

∫x².d/dx(x²) = ∫1dx
∫x²2xdx = ∫1dx
x⁴/2 + C = x where L.H.S. is not same as x³/3 + C

How is it possible?

 

[betel]

In order for the second step to be possible your function has to satisfy the first property. Which your example does not. Watch also that in the first line of your example you should have d(x^2) and not dx (x^2).

This is just the elaborate way of sovling the differential equation. Later you will do it without the explicit integrals.

 

[zorro]

Can you show me how to do it without integrals?

 

[betel]

You can rewrite the first equation as a total derivative
f\ f'=\frac{d}{dx}(f^2)=1
From this you immediately see, that f^2=x+C
Strictly speaking you still do the integral, but as they are so easy you can immediately do them in your head.

 

[zorro]

You can rewrite the first equation as a total derivative
f\ f'=\frac{d}{dx}(f^2)=1
From this you immediately see, that f^2=x+C

Is there a 2 missing in R.H.S. of first equation (numerator)?

 

[betel]

Yes, your are right.
<br /> f\ f&#039;=\frac{1}{2}\frac{d}{dx}(f^2)=1<br />