Question

The degree of the differential equation y = x ${\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dx}{dy}\right)}^2$ is _____________________.

Answer 1

Given: y = x ${\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dx}{dy}\right)}^2$


$$\begin{gathered} y=x{\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dx}{dy}\right)}^2 \\ \Rightarrow y=x{\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dy}{dx}\right)}^{-2} \\ \Rightarrow y=x{\left(\frac{dy}{dx}\right)}^2+\frac{1}{{\left({\displaystyle \frac{dy}{dx}}\right)}^2} \\ \Rightarrow y{\left(\frac{dy}{dx}\right)}^2=x{\left(\frac{dy}{dx}\right)}^2{\left(\frac{dy}{dx}\right)}^2+1 \\ \Rightarrow y{\left(\frac{dy}{dx}\right)}^2=x{\left(\frac{dy}{dx}\right)}^4+1 \end{gathered}$$

Here, power of highest order derivative is 4.

Hence, the degree of the differential equation y = x ${\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dx}{dy}\right)}^2$ is 4.