Second Order Linear

A linear homogeneous equation with constant coefficients is solved through its auxiliary equation, obtained by substituting the trial solution y = e^{mx}, which converts derivatives into powers of m. The justification is that exponential functions reproduce themselves under differentiation, so the differential equation becomes a polynomial equation in m. Replacing \frac{d^2y}{dx^2} with m^2, \frac{dy}{dx} with m, and y with 1 gives the characteristic equation m^2 - 5m + 6 = 0. Factoring yields (m-2)(m-3) = 0, so the roots are m = 2 and m = 3, two distinct real values. Distinct real roots give a general solution that is a linear combination of the corresponding exponentials: y = C_1 e^{2x} + C_2 e^{3x}. Option y = C_1 e^{-2x} + C_2 e^{-3x} would arise from roots -2 and -3, contradicting the middle coefficient sign. Option (C_1 + C_2 x)e^{2x} corresponds to a repeated root, which does not occur here. Option with cosine and sine requires complex roots, which the real discriminant rules out. This is the canonical JEE Advanced constant-coefficient method. As a final check, the discriminant 25 - 24 = 1 is positive, confirming two distinct real roots and validating the exponential form.