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√ D 2 Y D X 2 = √ D Y D X

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Question

\[\sqrt[3]{\frac{d^2 y}{d x^2}} = \sqrt{\frac{dy}{dx}}\]
Sum
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Solution

\[\sqrt[3]{\frac{d^2 y}{d x^2}} = \sqrt{\frac{dy}{dx}}\]

\[ \Rightarrow \left( \frac{d^2 y}{d x^2} \right)^\frac{1}{3} = \left( \frac{dy}{dx} \right)^\frac{1}{2} \]

Taking cubes of both the sides, we get

\[ \Rightarrow \frac{d^2 y}{d x^2} = \left( \frac{dy}{dx} \right)^\frac{3}{2} \]

Squaring both the sides, we get

\[ \Rightarrow \left( \frac{d^2 y}{d x^2} \right)^2 = \left( \frac{dy}{dx} \right)^3 \]

\[ \Rightarrow \left( \frac{d^2 y}{d x^2} \right)^2 - \left( \frac{dy}{dx} \right)^3 = 0\]

In this differential equation, the order of the highest order derivative is 2 and its power is 2. So, it is a differential equation of order 2 and degree 2.

Thus, it is a non-linear differential equation, as its degree is 2, which is greater than 1.

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Chapter 21: Differential Equations - Exercise 22.01 [Page 5]

APPEARS IN

R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 21 Differential Equations
Exercise 22.01 | Q 6 | Page 5

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