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1998

JEE Main 1998 Mathematics question on Differential Equations

The order of the differential equation whose general solution is given by y=(c1+c2)cos⁡(x+c3)−c4ex+c5y = (c_1 + c_2)\cos(x + c_3) - c_4 e^{x+c_5}, where c1,c2,c3,c4,c5c_1, c_2, c_3, c_4, c_5 are arbitrary constants, is
  1. A.
    5
  2. B.
    4
  3. C.
    3
  4. D.
    2
Show answer and solution

Correct option: C

Solution

The five constants are not all independent. c₁ and c₂ appear only as the sum (c₁ + c₂), so they merge into one constant A = c₁ + c₂. Similarly, c₄ and c₅ combine: c₄·eˣ⁺ᶜ⁵ = (c₄·ec₅)·eˣ = B·eˣ, a single constant B. This leaves three independent arbitrary constants: A, c₃, and B. The solution is effectively y = A·cos(x + c₃) − B·eˣ, which has exactly 3 independent constants, so the order of the differential equation is 3.

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