Find Two Differential Equations Whose Solution is y = A*e^x + B*cos(x)

In this video we have a solution of a differential equation y = Ae^x + Bcos(x). We are asked to find TWO differential equations who have this as a solution. This is a fun problem. I hope this helps someone.

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[ronitshah]

there's a simpler way to solve this, one diffeq is just W(e^x, cos(x), y)=0 where W is wronskian, cause if y is linearly dependent to e^x and cos(x) the wronskian would just be 0. For the second, you could do something as simple as take the derivative of all of the inputs to the wronskian, so W(e^x, sin(x), y')=0. Also, if the solution isn't in a linear form, so you can't use wronskian, there's still an easier way by isolated for each constant then taking a derivative. For example, rearrange to get A=e^-x(y-Bcos(x)) then take the derivative of both sides, which gets rid of A. After than isolate B and take another derivative, and you end up with a second order diffeq without any integration constants that has that as the only solution