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The solution of the differential equation k² d²y/dx² = y - y₂ under the boundary conditions (i) y = y₁ at x = 0 and (ii) y = y₂ at x = ∞, where k, y₁ and y₂ are constants, is

1. y = (y₁ - y₂) exp(-x/k²) + y₂
2. y = (y₂ - y₁) exp(-x/k) + y₁
3. y = (y₁ - y₂) sinh(x/k) + y₁
4. y = (y₁ - y₂) exp(-x/k) + y₂

Correct answer: y = (y₁ - y₂) exp(-x/k) + y₂

Solution

The correct option accurately reflects the solution to the differential equation, incorporating the exponential decay factor that satisfies the boundary conditions, where the function approaches y₂ as x approaches infinity and equals y₁ at x = 0.

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