Question

The general solution of the differential equation $$ (D^2 - 5D + 6) y = e^{3x} $$ is:

• A. \( c_1 e^{2x} + c_2 e^{3x} + e^{3x} \)
• B. \( c_1 e^{2x} + c_2 e^{3x} + x e^{3x} \)
• C. \( c_1 e^{-2x} + c_2 e^{-3x} + e^{3x} \)
• D. \( c_1 e^{-2x} + c_2 e^{-3x} + x e^{3x} \)

Hint:

When RHS is solution of homogeneous equation, multiply particular integral by \( x \).

Answer 1

The Correct Option is B

The characteristic equation: \[ m^2 - 5m + 6 = 0 \Rightarrow (m - 2)(m - 3) = 0 \] Roots: \( m = 2, 3 \) Particular integral for RHS \( e^{3x} \) since \( e^{3x} \) is solution of homogeneous equation, multiply by \( x \): \[ y_p = A x e^{3x} \] General solution: \[ y = c_1 e^{2x} + c_2 e^{3x} + x e^{3x} \]