1. **Problem Statement:** Solve the first order differential equation $\frac{dy}{dx} = y$. 2. **Formula and Explanation:** This is a separable differential equation. The general form is $\frac{dy}{dx} = ky$, where $k$ is a constant. The solution uses separation of variables: $$\frac{dy}{y} = dx$$ 3. **Step-by-step Solution:** - Integrate both sides: $$\int \frac{1}{y} dy = \int 1 dx$$ - This gives: $$\ln|y| = x + C$$ - Exponentiate both sides to solve for $y$: $$y = e^{x+C} = e^C e^x$$ - Let $A = e^C$, a constant, so: $$y = Ae^x$$ 4. **Explanation:** The solution shows that the function $y$ grows exponentially with $x$. The constant $A$ depends on initial conditions. **Final answer:** $$y = Ae^x$$