Q.Solve : $9y''-12y'+4y=0$

Equal roots $m=\tfrac23$ give $y=(A+Bx)e^{2x/3}$.

$9y''-12y'+4y=0.$

Step 1 — auxiliary (characteristic) equation. Replace $y''\to m^2,\ y'\to m,\ y\to1$:

$9m^{2}-12m+4=0.$

Step 2 — solve it.

$9m^{2}-12m+4=(3m-2)^{2}=0\ \Rightarrow\ m=\frac23,\ \frac23\ (\text{repeated root}).$

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