Question

If the solution curve of the differential equation \[ \frac{dy}{dx} = \frac{x + y - 2}{x - y} \] passing through the point \( (2, 1) \) is \[ \tan^{-1}\left(\frac{y - 1}{x - 1}\right) - \frac{1}{\beta} \log_e\left(\alpha + \left(\frac{y - 1}{x - 1}\right)^2\right) = \log_e |x - 1|, \] then \( 5\beta + \alpha \) is equal to

Answer 1

Correct Answer: 11

Given:
\(\frac{dy}{dx} = \frac{x + y - 2}{x - y}.\)

Substitute:
\(x = X + h, \quad y = Y + k.\)

Let:
\(h + k = 2, \quad h - k = 0 \implies h = k = 1.\)

So:
\(Y = vX, \quad \frac{dv}{dX} = \frac{1 + v^2}{1 - v}.\)

Integrating and applying the condition (2, 1):
\(\tan^{-1} \left(\frac{y - 1}{x - 1}\right) = \frac{1}{2} \log_e \left(1 + \left(\frac{y - 1}{x - 1}\right)^2\right) - \log_e |x - 1|.\)

From the equation:  
\(\alpha = 1, \quad \beta = 2.\)

Calculating \(5\beta + \alpha\):
\(5\beta + \alpha = 5 \times 2 + 1 = 11.\)

Thus, the Correct Answer is 11.

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The Correct answer is: 11