separation of variables

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 on both sides. $$\frac{dy}{g(y)}=f(x)dx$$ Without abusing Leibniz notation, the method actually makes use of the chain rule theorem and the fact that if two integrable functions are equivalent, then their primitives differ by a constant $C$ . Here, we write $y = y(x)$ and $\frac{dy}{dx} = y'(x)$ for clarity: $$\int \frac{y'(x)}{g(y(x))} dx = \int f(x) dx + C$$ By the chain rule of integration, also known as variable substitution or change of variable, the left hand side can be written as an integral in the variable $y$ : $$\int \frac{dy}{g(y)} = \int f(x) dx + C$$ 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\ $ $\frac{dP}{dt} = 0.15P$ $ The solution of this equation is relatively straightforward, we simply separate the variables algebraically and integrate. $ $\int \frac{dP}{P} = \int 0.15\;dt$ $ This is just $ 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 $ $P(t) = P_0e^{kt}$ $ \end{document} $