Washington 2.18d, elliptic curves change of variables in characteristic 3

Question

I got stuck on the Exercise 2.18d in the Book: Elliptic Curves: Number Theory and Cryptography. The problem is:

Show that if $y^2 = x^3 +a_4 x+a_6$ and $y^2 = x^3 +a'_4x^2 + a'_6$ are two elliptic curves (in characteristic 3), then there is a change of variables $ y \mapsto ay , x \mapsto bx +c$, with $a,b \in \overline{K}^{\times}$ and $c \in \overline{K}$, that changes one equation into the other.

My Problem is transforming the $x$ into the $x^2$, I guess you should somehow use that we operate in char 3. Since $(bx +c)^3 = b^3x^3 + c^3$ we can't get our $x^2$ from here.

Edit: It seems my version of the Book has a spelling mistake and the exercise should be $y² = x³ +a_4x +a⁶$ and $y²=x³ + a'_4x + a'_6$. I use the version "Elliptic Curves: Number Theory and Cryptography, 2nd Edition, CRC Press, Versiondate:20131121". If someone should stumble on the same problem.

Answer 1

This is a spelling mistake in the Book. I found an errata for the Book.

Show that if $y^2 = x^3 +a_4 x+a_6$ and $y^2 = x^3 +a'_4{\color{red}{x}} + a'_6$ are two elliptic curves (in characteristic 3), then there is a change of variables $ y \mapsto ay , x \mapsto bx +c$, with $a,b \in \overline{K}^{\times}$ and $c \in \overline{K}$, that changes one equation into the other.