This DE has characteristic equation
[tex]4r^2 - 12r + 9r = (2r - 3)^2 = 0[/tex]
with a repeated root at r = 3/2. Then the characteristic solution is
[tex]y_c = C_1 e^{\frac32 x} + C_2 x e^{\frac32 x}[/tex]
which has derivative
[tex]{y_c}' = \dfrac{3C_1}2 e^{\frac32 x} + \dfrac{3C_2}2 x e^{\frac32x} + C_2 e^{\frac32 x}[/tex]
Use the given initial conditions to solve for the constants:
[tex]y(0) = 3 \implies 3 = C_1[/tex]
[tex]y'(0) = \dfrac52 \implies \dfrac52 = \dfrac{3C_1}2 + C_2 \implies C_2 = -2[/tex]
and so the particular solution to the IVP is
[tex]\boxed{y(x) = 3 e^{\frac32 x} - 2 x e^{\frac32 x}}[/tex]
