$\frac{dy}{dx}=y$This is a separable ordinary differential equation, so$\frac{dy}{y}=dx$You now integrate both sides,$\int\limits_{}^{}\frac{dy}{y}=\int\limits_{}{}dx \rightarrow \ln(y)=x+c$Exponentiate both sides to solve for y (which is what you need),$e^{\ln(y)}=e^{x+c}$The left-hand side is just$e^{\ln(y)}=y$by definition of natural logarithm, while the right-hand side is just,$e^{x+c}=e^ce^x=Ae^x$because e^c is just a constant.Therefore$y=Ae^x$