If x = x(y) satisfies:

2y²(dx/dy) − 2xy + x² = 0

Given:

y = 1, x(e) = e

Find x(e²).

Given equation:

2y²(dx/dy) = 2xy − x²

Let:

x = 2yu

After substitution and separation of variables, integrating and applying the initial condition x(e)=e, we obtain:

x(y) = (2y²)/3

At y = e²:

x(e²) = 2e²/3

Final Answer

2e²/3