Obtain the differential equation by eliminating the arbitrary constants: y=Acos(log x)+Bsin(log x)

Exercise 6.2 · Q14

y=A\cos(\log x)+B\sin(\log x)

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$y=A\cos(\log x)+B\sin(\log x)$ has two arbitrary constants. With $t=\log x$ , $y$ solves $\dfrac{d^2y}{dt^2}+y=0$ (simple-harmonic form). Converting derivatives to $x$ : $x\dfrac{dy}{dx}=-A\sin t+B\cos t$ , and differentiating this again gives $\dfrac{dy}{dx}+x\dfrac{d^2y}{dx^2}=-\dfrac{1}{x}\left(A\cos t+B\sin t\right) …

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Q.Obtain the differential equation by eliminating the arbitrary constants: $y=A\cos(\log x)+B\sin(\log x)$

Concept understanding — Formation of a Differential Equation whose General Solution is Given

Q.Obtain the differential equation by eliminating the arbitrary constants: y=Acos⁡(log⁡x)+Bsin⁡(log⁡x)y=A\cos(\log x)+B\sin(\log x)y=Acos(logx)+Bsin(logx)

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Q.Obtain the differential equation by eliminating the arbitrary constants: $y=A\cos(\log x)+B\sin(\log x)$

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