The sum of the order and degree of the differential equation representing the family of curves $y = mx + m^4$, where m is arbitrary constant, is

The family of curves is given by $y = mx + m^4$, where $m$ is an arbitrary constant.

To find the differential equation, we need to eliminate the arbitrary constant $m$.

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Differentiating both sides with respect to $x$:

$\frac{dy}{dx} = m$

The derivative of $mx$ is $m$, and the derivative of $m^4$ is $0$ since $m$ is a constant.

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Substituting $m = \frac{dy}{dx}$ into the original equation $y = mx + m^4$:

$y = x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^4$

This is the required differential equation.

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The highest derivative present in the equation is $\frac{dy}{dx}$ (first derivative).

Order $= 1$

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The differential equation is polynomial in derivatives, and the highest power of $\frac{dy}{dx}$ is $4$.

Degree $= 2$

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$\text{Sum} = \text{Order} + \text{Degree}$

$\text{Sum} = 1 + 4$

$\text{Sum} = 5$

Therefore, the sum of the order and degree is $5$.