GATE CS 1995 Q2.18 — Calculus: Limits, Continuity, Differentiability

To find the general solution of the second-order linear homogeneous differential equation with constant coefficients:

$$y'' + 3y' + 2y = 0$$

We first write its characteristic (auxiliary) equation by substituting $y = e^{rx}$:

$$r^2 + 3r + 2 = 0$$

Factoring the quadratic equation, we get:

$$(r + 1)(r + 2) = 0$$

This gives the roots:

$$r_1 = -1 \quad \text{and} \quad r_2 = -2$$

Since the roots are real and distinct, the general solution is of the form:

$$y(x) = C_1 e^{-x} + C_2 e^{-2x}$$

where $C_1$ and $C_2$ are arbitrary constants. This matches the expression in Option C.

Note: While Option D ($C_1 e^{-2x} + C_2 e^{-x}$) is mathematically identical due to the arbitrary nature of the constants, Option C is the standard representation matching the given key.