How to Derive and Sketch the Frequency Response Function in an RC Circuit?

[discombobulated]

Homework Statement

The relationship between the input voltage V(t) and the output voltage across a resistor V_R(t) is:-

CR dV_R /dt + V_R = CR dV/dt

Circuit diagram of a capacitor and resistor with V_R(t) in series across V(t)

1. Show that the frequency response function G(iw) (w= omega) is given by G (iw) = iwCR/ (1 + iwCR)

2.Sketch the locus of the frequency response function, G(iw) (w= omega) on the argand diagram.

Homework Equations

CR dV_R /dt + V_R = CR dV/dt

G (iw) = iwCR/ (1 + iwCR)

The Attempt at a Solution

1. I got this G (iw) = iwCR/ (1 + iwCR) by voltage division, from the circuit, but how could I do that from the given equation?

2. By taking the real and imaginary parts; G(iw) = x + iy

x = (wcr)² /(1 + (wcr)²)
y = wcr/ ( 1 + (wcr)²)

G(iw) = (wcr)² /(1 + (wcr)²) + iwcr/ ( 1 + (wcr)²)

I'm having trouble eliminating wcr now...

 

[unplebeian]

I thought you'd apply Laplace transform and then subst s= jw to get the steady state solution. Then proceed to find the frequency response. But you arrived at the answer in one step, so why are you simplifying it further?