For each of the differential equations given below, indicate its order and degree (if defined).(i) `(d^2y)/(dx^2)+5x((dy)/(dx))^2-6x y=logx` (ii) `((dy)/(dx))^3-4((dy)/(dx))^2+7y=sinx`(iii) `(d^4y)/(dx^4)-sin((d^3y)/(dx^3))=0`

Explanation

`(i)` Given differential equation
`(d^(2)y)/(dx^(2))+5x((dy)/(dx))^(2)-6y=logx`
`implies (d^(2)y)/(dx^(2))+5x((dy)/(dx))^(2)-6y-logx=0`
The order with highest derivative in the given differential equation is `2`. Therefore the order of the differential equation is `2` and the degree of highest derivatives is `1`, so the degree of the differential equation is `1`.
`(ii)` Given differential equation
`((dy)/(dx))^(3)-4((dy)/(dx))^(2)+7y=sinx`
`implies((dy)/(dx))^(3)-4((dy)/(dx))^(2)+7y-sinx=0`
Here the derivative with maximum order is `((dy)/(dx))^(3`.
So its order is `1` and degree is `3`.
`(iii)` Given differential equation
`(d^(4)y)/(dx^(4))-sin((d^(3)y)/(dx^(3)))=0`.
The derivative with maximum order is `(d^(4)y)/(dx^(4))`, so the order of the differential equation is `4` but the degree is not defined because `sin((d^(3)y)/(dx^(3)))` is not a polynomial.