The solution of Differential equation $$ \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = e^{5x} $$ is -

Explanation

$$ \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = e^{5x} $$ Solution of Complementany Function D² - 3D + 2 = 0 D² - 2D - D + 2 = 0 D(D - 2) -1 (D - 2) = 0 (D - 2)(D - 1) = 0 D = 2, 1 Then, y = C₁eˣ + C₂e²ˣ Solution of Particular Integral P.I. = $\frac{1}{f(D)}$ e⁵ˣ = $\frac{1}{D^2 - 3D + 2}$ e⁵ˣ ($\because$ D = 5) = e²ˣ . $\frac{1}{25 - 15 + 2}$ = $\frac{1}{12}$ e⁵ˣ then general solution y = C₁eˣ + C₂e²ˣ + $\frac{1}{12}$ e⁵ˣ where C₁ C₁ & C₂ is a constant.