1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = y e^x$$ with the initial condition $$x=0, y=e$$. We need to find the value of $$y$$ when $$x=1$$. 2. **Rewrite the differential equation:** The equation is separable. We can write it as $$\frac{dy}{y} = e^x dx$$. 3. **Integrate both sides:** Integrate the left side with respect to $$y$$ and the right side with respect to $$x$$: $$\int \frac{1}{y} dy = \int e^x dx$$ This gives: $$\ln|y| = e^x + C$$ 4. **Solve for the constant $$C$$ using the initial condition:** When $$x=0$$, $$y=e$$, so: $$\ln e = e^0 + C \implies 1 = 1 + C \implies C=0$$ 5. **Write the explicit solution:** $$\ln|y| = e^x$$ Exponentiate both sides: $$y = e^{e^x}$$ 6. **Find $$y$$ when $$x=1$$:** $$y = e^{e^1} = e^e$$ **Final answer:** $$y = e^e$$ when $$x=1$$.