In my physics textbook there is a theorem that goes "If x1(t) and x2(t) are both solutions of a linear homogeneous d.e., then x(t) = x1(t) + x2(t) is also a solution."(adsbygoogle = window.adsbygoogle || []).push({});

I need to know if the inverse is true, i.e. If x(t) = x1(t) + x2(t) is a solution, does it implies that x1(t) and x2(t) are also solutions separately.

I tried to do a proof similar to that of the first theorem but I come to (for the case of a second order d.e. and using f(x) = g(x) + h(x) instead of x(t) = x1(t) + x2(t)):

[a2*d²g/dx² + a1*dg/dx + a0*g] + [a2*d²h/dx² + a1*dh/dx + a0*h] = 0

of course [a2*d²g/dx² + a1*dg/dx + a0*g] = 0 and [a2*d²h/dx² + a1*dh/dx + a0*h] = 0 is a solution but maybe it's [a2*d²g/dx² + a1*dg/dx + a0*g] = -[a2*d²h/dx² + a1*dh/dx + a0*h] too, right?

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# Is this theorem true?

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