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Why does the sum of two solutions remain a solution for a linear differential equation?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In mathematics, a linear differential equation is defined by the property that the unknown function and its derivatives appear only to the first power, with no cross-terms (such as the product of the function and its derivative).12

The reason the sum of two solutions remains a solution is rooted in the concept of a linear transformation. A linear differential equation can be viewed as an operator $T$ acting on a function $f$ . Because the equation is linear, this operator satisfies the principle of superposition: $T (u + v) = T (u) + T (v)$ . If $u$ and $v$ are both solutions to a homogeneous linear equation (where the equation equals zero), then $T (u) = 0$ and $T (v) = 0$ . Consequently, $T (u + v) = T (u) + T (v) = 0 + 0 = 0$ , confirming that their sum is also a solution.3

This property is fundamental to linear algebra, as it allows the set of solutions to form a vector space, where any linear combination of solutions is also a valid solution.3