Answer
See below
Work Step by Step
1. Determine Ly: $$Ly = y' - xy$$
(a) 2. Substitute y = $2e^{3x}$:
$$L(2e^{3x}) = (2e^{3x})^3- (2e^{3x})+4$$
3. Derivate:
$$L(2e^{3x}) = 54e^{3x} - 6e^{3x}+8e^{3x}$$
$$L(2e^{3x}) = 56e^{3x}$$
(b)
2. Substitute y = $3 ln(x)$:
$$L(3 ln(x)) = (3 ln(x))^3- (3 ln(x))+4$$
3. Derivate:
$$L(3 ln(x)) = \frac{6}{x^3} -\frac{3}{x}+12\ln x$$
(c) 2. Substitute y = $2e^{3x} + 3 ln(x)$:
$$L(2e^{3x} + 3 ln(x)) = (2e^{3x} + 3 ln(x))^3 - (2e^{3x} + 3 ln(x))+4$$
3. Derivate: $$L(2e^{3x} + 3 ln(x)) =54e^{3x}-6e^{3x}+8e^{3x}+\frac{6}{x^3}-\frac{3}{x}+12\ln x$$
$$L(2e^{3x} + 3 ln(x))=56e^{3x}+\frac{6}{x^3} -\frac{3}{x}+12\ln x$$
