Step By Step Calculus » 16.2 - Separable Differential Equations

show_hiddenExpand All [+]
Synopsis
A differential equation (DE) is an equation involving a function and one or more of its derivatives. The order of the highest derivative in a differential equation is the order of the equation.
Some differential equations come with some conditions such as the values of the function at certain values of xx. Such type of conditions are called initial conditions or boundary conditions, and the problem of solving such differential equations is known as initial-value problem or boundary-value problem.
Separable differential equations are the first-order differential equations that can be written in the form

\displaystyle { \textbf{(Separable DE)}\qquad\qquad \frac{dy}{dx}=\frac{f(x)}{g(y)}, g(y)\neq 0\quad\textrm{ or }\quad g(y)dy=f(x)dx. }\displaystyle { \textbf{(Separable DE)}\qquad\qquad \frac{dy}{dx}=\frac{f(x)}{g(y)}, g(y)\neq 0\quad\textrm{ or }\quad g(y)dy=f(x)dx. }

One can solve such equations by writing it as g(y)dy=f(x)dxg(y)dy=f(x)dx and then integrating both side of the equations, i.e.

\displaystyle { \textbf{(Integral Form for Separable DE)}\qquad\qquad \int g(y)dy=\int f(x)dx.}\displaystyle { \textbf{(Integral Form for Separable DE)}\qquad\qquad \int g(y)dy=\int f(x)dx.}