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Miscellaneous Exercise 6(II) · Q124

Q.Obtain the differential equation by eliminating the arbitrary constants: y=acos⁡(log⁡x)+bsin⁡(log⁡x)y=\sqrt{a\cos(\log x)+b\sin(\log x)}

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y=acos⁡(log⁡x)+bsin⁡(log⁡x)y=\sqrt{a\cos(\log x)+b\sin(\log x)}, i.e. y2=acos⁡(log⁡x)+bsin⁡(log⁡x)y^2=a\cos(\log x)+b\sin(\log x) — the same form as Acos⁡(log⁡x)+Bsin⁡(log⁡x)A\cos(\log x)+B\sin(\log x) but for z=y2z=y^2, so zz solves x2z′′+xz′+z=0x^2z''+xz'+z=0. With z=y2z=y^2, z′=2ydydxz'=2y\dfrac{dy}{dx}, z′′=2(dydx)2+2yd2ydx2z''=2\left(\dfrac{dy}{dx}\right)^2+2y\dfrac{d^2y}{dx^2}, substituting gives $2x^2\left(\dfrac{dy}{dx}\right)^2+2x^ …

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