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