$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}-3\frac{\mathrm{d}y}{\mathrm{d}x}+2y=0$$ How can I prove that $y=c_1e^x+c_2e^{2x}$ is the only solution (or most general solution) to this differential equation?
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Show that the differential equation can be written as $$ e^{2x}\frac{d}{dx}\left[e^{-x}\frac{d}{dx}(e^{-x}y)\right]=0. $$ This form of the equation can be integrated directly.
Disintegrating By Parts
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The most general solution to a 2nd order differential equation has 2 arbitrary constants and $e^x$ and $e^{2x}$ are linearly independent so they form a basis for all solutions. (Adding another function of $x$ that is independent of both $e^x$ and $e^{2x}$ would involve a third arbitrary constant.)
