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
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
There is one arbitrary constant, so one differentiation is enough to eliminate it:
\( \displaystyle \frac{dy}{dx} = m \)
Putting \(m = \frac{dy}{dx}\) back into \(y = mx + m^4\):
\( \displaystyle y = x\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^4 \)
The highest derivative is \(\frac{dy}{dx}\), so the order is 1. The equation is a polynomial in \(\frac{dy}{dx}\) and its highest power is 4, so the degree is 4.
\(\text{Order} + \text{degree} = 1 + 4 = 5\)