Separation of variables is a valuable tool for solving differential equations of the form $$\frac{dy}{dx}=f(x)g(y)$$ The above equation can be rearranged algebraically through Leibniz notation to separate the variables and be conveniently integrable. $$\frac{dy}{g(y)}=f(x)dx$$ It follows then that $$\int\frac{dy}{g(y)} = F(x) + C$$ where $F(x)$ is the antiderivative of $f$ and $C$ is a constant of integration. This gives a general form of the solution. An explicit form may be derived by an initial value.

Example: A population that is initially at $200$ organisms increases at a rate of $15\%$ each year. We then have a differential equation $$\frac{dP}{dt} = 0.15P$$ The solution of this equation is relatively straightforward, we simple separate the variables algebraically and integrate. $$\int \frac{dP}{P} = \int 0.15\;dt$$ This is just $\ln P = 0.15t + C$ or $$P=Ce^{0.15t}$$ When we substitute $P(0)=200$ , we see that $C=200$ . This is where we get the general relation of exponential growth $$P(t) = P_0e^{kt}$$

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