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