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(D) Given differential equation is $\left(\frac{d^2 y}{d x^2}\right)^{-\frac{7}{2}} \left(\frac{d^3 y}{d x^3}\right)^2 = \left(\frac{d^2 y}{d x^2}\rig

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$2$

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(D) Given differential equation is $\left(\frac{d^2 y}{d x^2}\right)^{-\frac{7}{2}} \left(\frac{d^3 y}{d x^3}\right)^2 = \left(\frac{d^2 y}{d x^2}\right)^{-\frac{5}{2}} \left(\frac{d^4 y}{d x^4}\right)$.Multiply both sides by $\left(\frac{d^2 y}{d x^2}\right)^{\frac{7}{2}}$ to eliminate negative exponents:$\left(\frac{d^3 y}{d x^3}\right)^2 = \left(\frac{d^2 y}{d x^2}\right) \left(\frac{d^4 y}{d x^4}\right)$.To remove the fractional power,we square both sides:$\left(\frac{d^3 y}{d x^3}\right)^4 = \left(\frac{d^2 y}{d x^2}\right)^2 \left(\frac{d^4 y}{d x^4}\right)^2$.The highest order derivative present is $\frac{d^4 y}{d x^4}$,so the order is $4$.The power of the highest order derivative is $2$,so the degree is $2$.The difference between order and degree is $4 - 2 = 2$.

The difference of the order and degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^{-\frac{7}{2}}\left(\frac{d^3 y}{d x^3}\right)^2-\left(\frac{d^2 y}{d x^2}\right)^{-\frac{5}{2}}\left(\frac{d^4 y}{d x^4}\right)=0$ is

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