Discussion
If \(y_{1}\) and \(y_{2}\) are two different solutions of the ordinary differential equation
\(y' + \sin(e^{x})y = \cos(e^{x+1}),\ 0 \le x \le 1,\)
then which one of the following is its general solution on [0, 1]?
Answer: B
The general solution is \(y_{1} + c(y_{1} - y_{2}),\ c \in \mathbb{R}\).
The equation is linear and first order but non-homogeneous. The difference \(y_{1} - y_{2}\) of two different solutions is a non-zero solution of the homogeneous equation, which spans its one-dimensional solution space. The general solution is a particular solution plus any multiple of that: \(y_{1} + c(y_{1} - y_{2})\).
\(c_{1}y_{1} + c_{2}y_{2}\) and \(c_{1}(y_{1} + y_{2}) + c_{2}(y_{1} - y_{2})\) are generally not solutions, because a linear combination of solutions of a non-homogeneous equation solves it only if the coefficients of \(y_{1}, y_{2}\) add to 1. \(cy_{1} + (y_{1} - y_{2})\) fails for the same reason unless \(c = 1\).
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