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Suppose $ y_1 y_2 $ is the product of the two solutions of the differential equation $ y'' + p_1 y' + p_2y = 0 $ and it is a constant. How can the requirement for this be $2p_1p_2 + p_2' = 0$? I have tried several manipulations but cannot get the desired requirement. Any hints?


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You didn't try enough. You should use the fact that constants have a zero derivative, differentiating over and over again, and using the differential equation to always get expressions involving only $y_1,y_2$ and their first derivatives. – Yuval Filmus Jun 30 '11 at 4:15
up vote 5 down vote accepted

Let $y_1 y_2 = C \neq 0$. Differentiate once: $$y'_1 y_2 + y_1 y'_2 = 0. $$ Differentiate again: $$y''_1 y_2 + 2y'_1y'_2 + y_1 y''_2 = 0. $$ Substitute the differential equation: $$2y'_1 y'_2 = p_1 y'_1 y_2 + p_2 y_1 y_2 + p_1 y_1 y'_2 + p_2 y_1 y_2 = 2Cp_2. $$ Differentiate this: $$2y''_1 y'_2 + 2y'_1 y''_2 = 2Cp'_2. $$ Substitute the differential equation: $$0 = 2Cp'_2 + 2p_1y'_1y'_2 + 2p_2y_1y'_2 + 2p_1y'_1y'_2 + 2p_2y'_1y_2 = 2Cp'_2 + 4p_1y'_1y'_2 = 2Cp'_2 + 4Cp_1p_2. $$ Since $C \neq 0$, we deduce that $$ p'_2 + 2p_1 p_2 = 0. $$

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