(2.2) Solve 2nd Order Linear Homogeneous ODEs with Constant Coefficients

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This video explains how to solve second order linear homogeneous differential equations with constant coefficients.
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can you explain a more in depth proof of how e^2x and e^4x are linearly independent and how we can write the general solution as a linear combination of them after knowing they are linearly independent?

joeahmadguitar