Question

\[\frac{d^4 y}{d x^4} = \left\{ c + \left( \frac{dy}{dx} \right)^2 \right\}^{3/2}\]

Answer 1

\[\frac{d^4 y}{d x^4} = \left[ c + \left( \frac{dy}{dx} \right)^2 \right]^\frac{3}{2} \]
Squaring both sides, we get
\[ \Rightarrow \left( \frac{d^4 y}{d x^4} \right)^2 = \left[ c + \left( \frac{dy}{dx} \right)^2 \right]^3 \]
\[ \Rightarrow \left( \frac{d^4 y}{d x^4} \right)^2 = c^3 + 3 c^2 \left( \frac{dy}{dx} \right)^2 + 3c \left( \frac{dy}{dx} \right)^4 + \left( \frac{dy}{dx} \right)^6\]
In this differential equation, the order of the highest order derivative is 4 and its power is 2. So, it is a differential equation of order 4 and degree 2.
Thus, it is a non-linear differential equation, as its degree is 2, which is greater than 1.