How did they find the differential operator in this problem?

Correct choice: (g) y_p = A x*2 + Bx + C x e\"{4x} + D e\"{7x} Step-by-step: 1. The differential operator is L(D) = D*3 - 9D*2 + 20D = D(D - 4)(D - 5). Hence the homogeneous solution contains e\"{Ox}=1, e\"{4x}, e\"{5x}. 2. Right-hand side terms: Polynomial x (degree 1). Since r = O is a root of the characteristic (multiplicity 1), multiply the usual degree-1 polynomial trial (Ax + B) by x Ax*2 + Bx. * Term 5 e\"{4x}. Because r = 4 is a root (multiplicity 1), multiply the trial Ce*{4x} by x C x e*{4x}. Term 9 e\"{7x}. Since r = 7 is not a root, use De\"{7x}. 3. Combine all contributions: y_p = Ax*2 + Bx + C x e\"{4x} + D e\"{7x}. Thus, option (g) is the correct form for a particular solution