Question

If the solution curve f(x, y) = 0 of the differential equation (1+log_e x) \(\frac{dx}{dy} \)- x log_e x = e^y , x > 0, passes through the points (1,0) and (α, 2) then α^a is equal to

• A. \(e^{e^{2}}\)
• B. \(e^{\sqrt2e^{2}}\)
• C. \(e^{2e^{\sqrt2}}\)
• D. \(e^{2e^{2}}\)

Answer 1

The Correct Option is D

The given differential equation is: \[ (1 + \log x) \frac{dx}{dy} - x \log x = e^y \] Let \( x \log x = t \). Then, \[ (1 + \log x) \frac{dx}{dy} = t \] Now, we integrate both sides to find \(t = y + c\). Using the given points, we find: \[ \alpha^4 = e^{2e^2} \]