Prove an ODE is linear if it can be solved by combining any two solutions.
I'm trying to prove that for any ordinary differential equation, f(t, x, ... , x^(n)) = 0, that the equation must be linear if for any two solutions, x1 and x2, ax1+bx2 is also a solution (for constant a and b). I've been thinking about this for a while, and I don't know where to start wtih it.
I've already shown the reverse, that if the equation is linear that a linear combination of any two solutions will also be a solution, which I did by writing the equation as a sum and putting in ax1+bx2, expanding it and showing it was still equal to zero.
Any help would be much appreciated, thank you.
I've already shown the reverse, that if the equation is linear that a linear combination of any two solutions will also be a solution, which I did by writing the equation as a sum and putting in ax1+bx2, expanding it and showing it was still equal to zero.
Any help would be much appreciated, thank you.
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