Verify: $y=3\cos(\log x)+4\sin(\log x)$, and $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$

Verify: y=3cos(log x)+4sin(log x), and x²(d²y)/(dx²)+x(dy)/(dx)+y=0

Miscellaneous Exercise 6(II) · Q118

y=3\cos(\log x)+4\sin(\log x)

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$y=3\cos(\log x)+4\sin(\log x)$ has the same functional form as $A\cos(\log x)+B\sin(\log x)$ , which (as derived in this chapter) solves $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$ — checked directly by computing $y'=\dfrac{-3\sin(\log x)+4\cos(\log x)}{x}$ and $y''$ , and confirming the com …

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Q.Verify: $y=3\cos(\log x)+4\sin(\log x)$ , and $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$

Concept understanding — General and Particular Solutions of a Differential Equation

Q.Verify: y=3cos⁡(log⁡x)+4sin⁡(log⁡x)y=3\cos(\log x)+4\sin(\log x)y=3cos(logx)+4sin(logx), and x2d2ydx2+xdydx+y=0x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0x2dx2d2y​+xdxdy​+y=0

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Q.Verify: $y=3\cos(\log x)+4\sin(\log x)$ , and $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$

Unlock everything free for 14 days