If `y=3cos(logx)+4sin(logx),\ ` then show that `x^2dot(d^2\ y)/(dx^2)+(dy)/(dx)+y=0`

Explanation

`y = 3cos(logx)+4sin(logx)`

Let `logx = t =>1/xdx = dt=>dt/dx = 1/x`

Then, `y = 3cost+4sint`

`=>dy/dt = -3sint+4cost`

`:.dy/dx = dy/dt*dt/dx = (-3sint+4cost)1/x`

`=>xdy/dx = 4cost-3sint`

Differentiating w.r.t. `x`,

`=>x(d^2y)/dx^2+dy/dx = (-4sint-3cost)dt/dx`

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