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If y1 and y2 are two particular solutions of the first-order linear differential equation dy/dx + P(x)*y = Q(x), which of the following are true? (A) y = y1 + C*(y1 - y2) is the general solution for arbitrary constant C. (B) y = y1 + C*(y1 + y2) is the general solution for arbitrary constant C. (C) alpha*y1 + beta*y2 is also a solution whenever alpha + beta = 1. (D) alpha*y1 + beta*y2 is also a solution whenever alpha - beta = 1.

1. (A) y = y1 + C*(y1 - y2) is the general solution.
2. (B) y = y1 + C*(y1 + y2) is the general solution.
3. (C) alpha*y1 + beta*y2 is a solution if alpha + beta = 1.
4. (D) alpha*y1 + beta*y2 is a solution if alpha - beta = 1.

Correct answer: (A) y = y1 + C*(y1 - y2) is the general solution.

Solution

Since y1 and y2 are both solutions of the linear DE, their difference (y1-y2) satisfies the homogeneous equation dy/dx + Py = 0. The general solution is any particular solution plus an arbitrary multiple of the homogeneous solution: y = y1 + C*(y1-y2). Also, alpha*y1+beta*y2 satisfies the DE iff alpha+beta=1 (the DE is affine, not linear).

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