GATE 2021 CH – Question 46
For the ordinary differential equation $\frac{d^3y}{dt^3}+6\frac{d^2y}{dt^2}+11\frac{dy}{dt}+6y=1$, with initial conditions $y(0)=y\prime(0)=y^{\prime\prime}(0)=y^{\prime\prime\prime}(0)=0$, the value of $\lim_{t\to\infty}y(t)$ is _____ (round off to 3 decimal places).
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Correct answer: 0.157 to 0.177
Explanation
At a settled constant output the derivative terms vanish, so the forced equilibrium is $6y_\infty=1$, giving $y_\infty=1/6$. Check that transients decay: the characteristic polynomial factors as $s^3+6s^2+11s+6=(s+1)(s+2)(s+3)$, with all roots negative.
Equivalently, for zero initial displacement, velocity and acceleration, $Y(s)=1/[s(s+1)(s+2)(s+3)]$ and the final-value theorem gives $1/6$. **Answer: 0.167.**
The printed additional condition on the third derivative is inconsistent with the forcing at the initial instant; it does not alter the intended stable long-time value.