Let y=y(t) be a solution of the differential equation dy/dt + α y = γ e^(- β t) where, α > 0,β > 0 and γ > 0. Then lim _(t → ∞) y(t)

Solution

Maths sir · 4 steps

Multiply by e^(α t), solve, and watch every term carry a decaying exponential that pulls y to zero.

1. Integrating factor. This is linear in y with P = α, so the integrating factor is e^(α t). Multiplying through gives d/dt(y e^(α t)) = γ e^((α - β)t).
2. Case alpha not beta. Integrating gives y e^(α t) = γ/(α - β) e^((α - β)t) + C. Dividing by e^(α t) gives the solution below.

   y(t) = γ/(α - β) e^(-β t) + C e^(-α t)

3. Case alpha equals beta. Now the right side is the constant γ, so y e^(α t) = γ t + C and y = (γ t + C)e^(-α t). An exponential beats any power of t, so t e^(-α t) → 0.
4. So. Since α > 0 and β > 0, both e^(-α t) and e^(-β t) go to 0, whatever the value of C. In both cases lim(t → ∞) y(t) = 0, which is option (a).

The idea to keep. For a limit of a linear DE solution, solve it fully and check each term's exponent. Remember the special case where the exponents match: it brings a factor of t, but e⁻ᵏᵗ still wins.

Answer from the NTA final answer key. Solution by PrepWiser.