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If y=3cos(logx)+4sin(logx),show that x^...

If `y=3cos(logx)+4sin(logx),`show that `x^2y_2+x y_1+y=0`.

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`y=3" cos"(log x)+4" sin"(log x) " "`...(1)
`impliesy_(1)=(d)/(dx)[3" cos"(log x) +4 sin (log x)]`
`= -(3" sin"(log x))/(x)+(4" cos"(log x))/(x) `
`impliesxy_(1)= -3 "sin"(log x)+4cos(log x)`
Again differentiate both sides w.r.t.x
`x*y_(2)+y_(1)*1=(3cos(logx))/(x)-(4sin(log x))/(x)`
`implies x^(2)y_(2)+xy_(1)= -[3cos(logx)+4sin(logx)]= -y`
` " " `from equation (1)
`implies x^(2)y_(2)+xy_(1)+y=0 " " ` Hence proved.
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