For each of the differential equations given below, indicate its order and degree (if defined).

(i) $\frac{{{d^2}y}}{{d{x^2}}} + 5x{\left( {\frac{{dy}}{{dx}}} \right)^2} - 6y = \log x$

(ii) ${\left( {\frac{{dy}}{{dx}}} \right)^3} - 4{\left( {\frac{{dy}}{{dx}}} \right)^2} + 7y = \sin x$

(iii) $\frac{{{d^4}y}}{{d{x^4}}} - \sin \left( {\frac{{{d^3}y}}{{d{x^3}}}} \right) = 0$

(i) The differential equation is given as:
$\frac{{{d^2}y}}{{d{x^2}}} + 5x{\left( {\frac{{dy}}{{dx}}} \right)^2} - 6y = \log x$
$\Rightarrow$ $\frac{{{d^2}y}}{{d{x^2}}} + 5x{\left( {\frac{{dy}}{{dx}}} \right)^2} - 6y - \log x = 0$

The highest order derivative present in the differential equation is $\frac{{{d^2}y}}{{d{x^2}}}$.

Thus, its order is two. The highest power raised to $\frac{{{d^2}y}}{{d{x^2}}}$ is one. Hence, its degree is one.

(ii) The differential equation is given as:
${\left( {\frac{{dy}}{{dx}}} \right)^3} - 4{\left( {\frac{{dy}}{{dx}}} \right)^2} + 7y = \sin x$

$\Rightarrow$ ${\left( {\frac{{dy}}{{dx}}} \right)^3} - 4{\left( {\frac{{dy}}{{dx}}} \right)^2} + 7y - \sin x = 0$

The highest order derivative present in the differential equation is $\frac{{dy}}{{dx}}$. Thus, its order is one. The highest power raised to $\frac{{dy}}{{dx}}$ is three.

Hence, its degree is three.

(iii) The differential equation is given as:
$\frac{{{d^4}y}}{{d{x^4}}} - \sin \left( {\frac{{{d^3}y}}{{d{x^3}}}} \right) = 0$

The highest order derivative present in the differential equation is $\frac{{{d^4}y}}{{d{x^4}}}$. Thus, its order is four.

However, the given differential equation is not a polynomial equation. Hence, its degree is not defined.