The differential equation of the family of cu | vedclass

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(D) Given the family of curves: $y = ax + \frac{1}{a}$  $(i)$ Differentiating with respect to $x$, we get: $\frac{dy}{dx} = a$ Substituting $a = \frac

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$2$

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(D) Given the family of curves: $y = ax + \frac{1}{a}$  $(i)$ Differentiating with respect to $x$, we get: $\frac{dy}{dx} = a$ Substituting $a = \frac{dy}{dx}$ into equation $(i)$: $y = x \left(\frac{dy}{dx}\right) + \frac{1}{\frac{dy}{dx}}$ Multiplying both sides by $\frac{dy}{dx}$ to eliminate the fraction: $y \left(\frac{dy}{dx}\right) = x \left(\frac{dy}{dx}\right)^2 + 1$ The highest power of the highest order derivative $\frac{dy}{dx}$ is $2$. Therefore, the degree of the differential equation is $2$.

The differential equation of the family of curves $y = ax + \frac{1}{a}$, where $a (\neq 0)$ is an arbitrary constant, has the degree

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