Question

The degree and order of the differential equation $y=px+x\sqrt[3]{a^2{p}^2+b^2}$, where $p=\frac{dy}{dx}$ are respectively

• A. $3,1$
• B. $1,3$
• C. $1,1$
• D. $3,3$

Answer 1

The correct option is A

$3,1$

Explanation for the correct option:

To find degree and order:

Given $y=px+x\sqrt[3]{a^2{p}^2+b^2}$where $p=\frac{dy}{dx}$

$$\begin{gathered} \Rightarrow y=\frac{dy}{dx}x+x{\left(a^2{\left(\frac{dy}{dx}\right)}^2+b^2\right)}^{\frac{1}{3}} \\ \Rightarrow y-\frac{dy}{dx}x=x{\left(a^2{\left(\frac{dy}{dx}\right)}^2+b^2\right)}^{\frac{1}{3}} \\ \Rightarrow {\left(y-\frac{dy}{dx}x\right)}^3=x^3\left(a^2{\left(\frac{dy}{dx}\right)}^2+b^2\right) \\ \Rightarrow y^3-3y^2\frac{dy}{dx}+3y{\left(\frac{dy}{dx}\right)}^2-{\left(\frac{dy}{dx}\right)}^3=x^3\left(a^2{\left(\frac{dy}{dx}\right)}^2+b^2\right)\left[{\left(a-b\right)}^3=a^3-3a^{2}b+3a{b}^2-b^3\right] \end{gathered}$$

Order:

The highest order derivative $1$.

Power:

The power of the highest order derivative is $3$

Hence degree of the equation is $3$ and order is $1$.

Hence, option (A) is the correct.