The variables x and y satisfy the differential equation \(\frac{dy}{dx} = ky^3 e^{-x}\), where \(k\) is a constant. It is given that \(y = 1\) when \(x = 0\), and that \(y = \sqrt{e}\) when \(x = 1\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).