Diff Eq Y Equals Y 86Fd3A

1. **State the problem:** We need to solve the differential equation $$y' = y$$ where $y'$ denotes the derivative of $y$ with respect to $x$. 2. **Recall the formula:** This is a first-order linear differential equation and can be solved by separation of variables or recognizing it as an exponential growth/decay equation. 3. **Separate variables:** Write the equation as $$\frac{dy}{dx} = y$$ which implies $$\frac{dy}{y} = dx$$. 4. **Integrate both sides:** Integrate $$\int \frac{1}{y} dy = \int 1 dx$$ to get $$\ln|y| = x + C$$ where $C$ is the constant of integration. 5. **Solve for $y$:** Exponentiate both sides to remove the logarithm: $$|y| = e^{x+C} = e^C e^x$$ Let $A = e^C$, which is an arbitrary positive constant, so $$y = A e^x$$. 6. **General solution:** The solution to the differential equation is $$y = Ce^x$$ where $C$ is any real constant. This function satisfies the original differential equation because its derivative is itself multiplied by 1. **Final answer:** $$y = Ce^x$$