Answer
See below
Work Step by Step
1. Determine Ly: $$Ly = y' - xy$$ (a) 2. Substitute y = $2e^{3x}$:
$$L(2e^{3x}) = (2e^{3x})^2 - x^2(2e^{3x})+x$$
3. Derivate:
$$L(2e^{3x}) = 18e^{3x} - 6x^2e^{3x} +2xe^{3x}$$
(b)
2. Substitute y = $3 ln(x)$:
$$L(3 ln(x)) = (3 ln(x))^2- x^2(3 ln(x))+x$$
3. Derivate:
$$L(3 ln(x)) = -\frac{3}{x^2} -3x + 3xln(x) $$
(c) 2. Substitute y = $2e^{3x} + 3 ln(x)$:
$$L(2e^{3x} + 3 ln(x)) = (2e^{3x} + 3 ln(x))^2 - x^2(2e^{3x} + 3 ln(x))+x$$
3. Derivate: $$L(2e^{3x} + 3 ln(x)) =18e^{3x}-6x^2e^{3x}+2xe^{3x}-\frac{3}{x^2}-3x+3x \ln x$$
