Integral of x*delta((x/y)-t) dx from 0 to infinity

[Applejacks01]

Homework Statement

∫x*delta((x/y)-t) dx from 0 to infinity

Homework Equations

∫x*delta((x/y)-t) dx from 0 to infinity = ty*|y|*θ(ty)

The Attempt at a Solution

Okay, so using the transformation of variables technique via the Jacobian, I see where the |y| comes from. However, using the dirac delta method I have NO clue how that |y| is derived logically. I know that the property of the dirac delta is that, for ex, ∫f(x)*delta(x-a) dx from x = -infinity to infinity = f(a). In other words, we solve the equation x-a = 0 for x. Likewise, we have ((x/y)-t) = 0, solved for x = ty.

So I can see where ty*θ(ty) comes from...but how is the |y| derived?


Thank you very much.

 

[vela]

Try using the substitution u=x/y to get the argument of the delta function to look like it appears in the property you cited.

 

[Dick]

Try using the substitution u=x/y to get the argument of the delta function to look like it appears in the property you cited.

Don't forget that if y is negative you need to adjust the limits in the u integral. That's where the absolute value will come from.

 

[Applejacks01]

Ok assuming y is positive( it actually is for the question I was really working on)...
okay so u = x/y implies x = y*u. So we have (y*u*Delta(u-t))
we need to convert dx to du
x = y* u
Dx = y du
int(y*u*Delta(u-t)*ydu)
So u = t and we get
y*t*y*Heaviside(yt)




I can see how the y is derived now. Hopefully my logic is perfect in my derivation. Thank you!