separation of variables SeparationOfVariables 2002-02-26 00:09:44 2002-02-26 00:14:25 Algorithm % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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. \textbf{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 $$P(t) = P_0e^{kt}$$ [more later]