y alin (loger) Prove that er yat ry, ty=0

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( underline{Q}=y=a sin (ln x) ) ( begin{aligned} frac{d y}{d x} &=operatorname{acos}(ln x) cdot frac{1}{x}=frac{9}{x} cdot cos (ln x) frac{d^{2} y}{d x^{2}} &=frac{-a}{x^{2}} cos (ln x)+frac{a}{x} sin (ln x) cdot frac{1}{x} &=frac{-a}{x^{2}} cos left(ln (x)-frac{9}{x^{2}} sin (ln x)-(2)right.end{aligned} ) ( Rightarrow ) multiply 0 by ( x^{prime} ) & (2) by ( x^{2}, ) woget ( x y_{1}=a cos (ln x) ) ( x^{2} y_{2}=-a cos (ln x)-a sin (ln x) ) ( y=a sin (ln x)-(6) ) Ad of ( (9+sqrt{5}+6), ) we get ( x^{2} y_{2}+x y_{1}+y=0 )

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