The order of the differential equation of the family of curves \(y = Ae^{x + B} + Cx\), where A, B and C are arbitrary constants, is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
The order equals the number of essential (independent) arbitrary constants. Constants that can be combined into one count as one.
Using \(e^{x + B} = e^{x} \cdot e^{B}\):
\(y = (Ae^{B})\,e^{x} + Cx\)
A and B occur only through the single constant \(K = Ae^{B}\):
\(y = Ke^{x} + Cx\)
So the family has only 2 independent constants, K and C, and its differential equation is of order 2.
Why the other options are incorrect
Option (1) 3 — Incorrect. This counts A, B and C separately, but A and B combine into the single constant \(Ae^{B}\).
Option (3) 1 — Incorrect. The constants \(Ae^{B}\) and C multiply different functions, \(e^{x}\) and x, so they cannot be combined further.
Option (4) 4 — Incorrect. The family has at most three written constants, and only two independent ones.