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In textbooks, it's often casually mentioned, without explanation, that any two solutions added together is the general solution, the form of every other solution.

I don't understand why this is or where that idea comes from. Can anyone explain why this is so?

EDIT: This question was badly put. I'm going to post another question that makes more sense. Thanks for the feedback.

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Sometimes the general solution has the form $(C_0+C_1x)e^{\alpha x}$. – Hagen von Eitzen Nov 25 '12 at 23:37

It's not true that every solution has the form given in the title of the question.

It is true that any linear combination of solutions is a solution or, to put the same thing another way, that the set of all solutions forms a vector space. Do you need help proving that the set of all solutions forms a vector space?

The only question, then, is why the vector space has dimension 2. Well, one proves a theorem stating that there is a unique solution given the initial conditions $y(0)$ and $y'(0)$, and that settles it.

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The premise in your title is incorrect, even for constant-coefficient equations (and certainly not for non-constant-coefficient ones).
For example, $y'' + 2 y' + y = 0$ has the solutions $y = C_0 e^{-x} + C_1 x e^{-x}$.

What is true is that by the Existence and Uniqueness Theorem and linearity, the space of solutions of a homogeneous second-order linear differential equation is a two-dimensional vector space. So if you have two linearly independent solutions, all solutions are linear combinations of these two.

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While excellent answers have already been given, I think you might like some more concrete steps. Say I have two solutions, $y_1$ and $y_2$. If the equation is represented by some linear differential operator $D$ (it could be $D=\frac{d^2}{dx^2}+x$, $D=x^2\frac{d^2}{dx^2}-\frac{d}{dx}$ or any number of things), then

$$D(y_1)=0$$ $$D(y_2)=0$$

these are zero precisely because $y_1$ and $y_2$ are both solutions to the homogeneous ODE by assumption. So clearly


and if the operator is linear (which it is here)


So if you have two solutions, by linearity their sum is a solution as well. At its core this is just a fancy application of $0+0=0$. A very similar statement can be made about scalar multiples. Why two independent solutions are needed to find the rest (i.e, they form a basis) has been answered excellently by others.

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