Question

The order and degree of differential equation ${\left(\frac{d^{3}y}{d{x}^3}\right)}^{\frac{4}{5}}-2\left(\frac{dy}{dx}\right){\left(\frac{d^{2}y}{d{x}^2}\right)}^2=0$ are

• A. $2,10$
• B. $3,10$
• C. $3,5$
• D. $3,4$

Answer 1

The correct option is D

$3,4$

Explanation for the correct option

The order of a differential equation is defined as the highest order of the derivative.

The degree of the differential equation is defined as the power of the highest order of the derivative.

Given differential equation is,

${\left(\frac{d^{3}y}{d{x}^3}\right)}^{\frac{4}{5}}-2\left(\frac{dy}{dx}\right){\left(\frac{d^{2}y}{d{x}^2}\right)}^2=0$

$\Rightarrow {\left(\frac{d^{3}y}{d{x}^3}\right)}^{\frac{4}{5}}=2\left(\frac{dy}{dx}\right){\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

Now both side raise to the power of $5$

$$\begin{gathered} \Rightarrow {\left({\left(\frac{d^{3}y}{d{x}^3}\right)}^{\frac{4}{5}}\right)}^5={\left(2\left(\frac{dy}{dx}\right){\left(\frac{d^{2}y}{d{x}^2}\right)}^2\right)}^5 \\ \Rightarrow {\left(\frac{d^{3}y}{d{x}^3}\right)}^4=2^5{\left(\frac{dy}{dx}\right)}^5{\left(\frac{d^{2}y}{d{x}^2}\right)}^{10} \end{gathered}$$

Therefore Order$=3$, Degree $=4$

Hence, the correct option is $\left(\mathrm{D}\right)$.