Discussion
Consider a differential equation
\[ \frac{d^{2}x}{dt^{2}} + 2\frac{dx}{dt} + x = 0 \]
At time t = 0, it is given that x = 1 and \(\frac{dx}{dt}\) = 0. At t = 1, the value of x is :
Answer: B
At t = 1, x = 2/e.
The characteristic equation is \(m^{2} + 2m + 1 = 0\), with the repeated root \(m = -1\) (critical damping). So the general solution and the conditions give:
\[ \begin{aligned} x &= (A + Bt)e^{-t} \\ x(0) &= A = 1 \\ \dot{x}(0) &= B - A = 0 \Rightarrow B = 1 \end{aligned} \]
Hence \(x = (1 + t)e^{-t}\), and \(x(1) = 2e^{-1} = 2/e\). The option 1/e is what you get by dropping the \(Bt\) term that a repeated root requires.
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