Discussion
Consider the differential equation \(\frac{d^{2}y}{dx^{2}} - 2\frac{dy}{dx} + y = 0\).
The boundary conditions are \(y = 0\) and \(\frac{dy}{dx} = 1\) at \(x = 0\).
Then the value of \(y\) at \(x = 1/2\) is
Answer: C
The value is \(\frac{\sqrt{e}}{2}\).
The characteristic equation \(m^{2} - 2m + 1 = 0\) has the repeated root m = 1, so the general solution is \(y = (A + Bx)e^{x}\).
\[ \begin{aligned} y(0) = 0 &\Rightarrow A = 0 \\ y'(x) = Be^{x}(1 + x) &\Rightarrow y'(0) = B = 1 \\ y &= xe^{x} \end{aligned} \]
At x = 1/2, \(y = \frac{1}{2}e^{1/2} = \frac{\sqrt{e}}{2}\). \(\sqrt{e}\) leaves out the factor x = 1/2, and \(\sqrt{e/2}\) puts the 1/2 under the root. Zero would mean the trivial solution, which the condition y'(0) = 1 rules out.
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