If \(y = y(x)\), \(x \gt 0\), is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{y - 1}{x^2}\) satisfying \(y(1) = 2\), then \(y\left(\displaystyle\frac{1}{2}\right)\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(1 + \displaystyle\frac{1}{e}\)
The equation is variables separable (y stays greater than 1, since y(1) = 2):
\( \displaystyle \frac{dy}{y - 1} = \frac{dx}{x^2} \)
Integrating both sides, with \(\int x^{-2}\,dx = -\frac{1}{x}\):
\( \displaystyle \log_e (y - 1) = -\frac{1}{x} + C \)
Putting x = 1, y = 2:
\(0 = -1 + C\)
\(C = 1\)
\( \displaystyle \log_e (y - 1) = 1 - \frac{1}{x} \)
At \(x = \frac{1}{2}\), where \(\frac{1}{x} = 2\):
\(\log_e (y - 1) = -1\)
\( \displaystyle y = 1 + e^{-1} = 1 + \frac{1}{e} \)
Why the other options are incorrect
Option (1) \(1 + e\) — Incorrect. This comes from \(\int x^{-2}\,dx = +\frac{1}{x}\), a sign slip.
Option (2) \(\displaystyle\frac{1}{e}\) — Incorrect. This is the value of \(y - 1\), not of y.
Option (3) 2 — Incorrect. This is y(1), the given value; y changes between x = 1 and x = 1/2.