Question

Write the degree of the differential equation $\frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2=2x^2\log \left(\frac{d^{2}y}{d{x}^2}\right)$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2=2x^2\log \left(\frac{d^{2}y}{d{x}^2}\right) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2-2x^2\log \left(\frac{d^{2}y}{d{x}^2}\right)=0 \\ \mathrm{Here},\mathrm{we}\mathrm{observe}\mathrm{that}\mathrm{LHS}\mathrm{of}\mathrm{the}\mathrm{differential}\mathrm{equation}\mathrm{cannot}\mathrm{be}\mathrm{expressed}\mathrm{as}\mathrm{a}\mathrm{polynomial}\mathrm{in}\frac{dy}{dx}.\mathrm{So},\mathrm{its}\mathrm{degree}\mathrm{is}\mathrm{not}\mathrm{defined}. \end{gathered}$$