Connecting KdV to MKdV: Miura Transformation

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Homework Statement

Show how the Miura transformation u = -(v² + v_x) connects the KdV u_t + 6uu_x + u_xxx = 0 to the modified KdV v_t - 6v²v_x + v_xxx = 0
Which of the following is true, and why?
a) u solves the KdV if v solves the MKdV
b) v solves the MKdV if u solves the KdV
c) u solves the KdV if and only if v solves the MKdV.

Homework Equations

The Attempt at a Solution

Let u = -(v² + v_x) therefore u_t + 6uu_x + u_xxx = v_t + 6(-(v² + v_x))(2v+d/dx) + v_xxx = 0, I'm lost after this bit, any ideas?
I know a) is true because of application to the Miura transformation.

 

[Nino MMP]

b) is false as it does not hold for all solutions. c) is true, as u solves the KdV if and only if v solves the MKdV.