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

Question:

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

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

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

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

Solution:

(i) The differential equation is given as:

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

$\Rightarrow \frac{d^{2} y}{d x^{2}}+5 x\left(\frac{d y}{d x}\right)^{2}-6 y-\log x=0$

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:

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

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

The highest order derivative present in the differential equation is $\frac{d y}{d x}$. Thus, its order is one. The highest power raised to $\frac{d y}{d x}$ is three. Hence, its degree is three.

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

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.

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