A method of solving a differential equation by separating the functions of a variable with respective differential on one side and the functions of another variable with corresponding differential on other side of the equation is called the variables separable method of differential equations.

Let $x$ and $y$ represent two variables, then $dx$ and $dy$ are corresponding differentials. When the functions in $x$ and $y$ are denoted by $f(x)$ and $g(y)$ mathematically and form a differential equation in terms of them. The differential equation can be solved by the separation of variables to keep functions in $x$ with respective differential $dx$ on one side and to move the functions in $y$ with corresponding differential $dy$ on other side of the equation as follows.

$g(y)dy \,=\, f(x)dx$

Now, integrate both sides of the differential for solving the differential equation by variable separation.

$\implies$ $\displaystyle \int{g(y) \,}dy \,=\, \int{f(x) \,}dx+c$

Now, let us learn how to use the separation of variables method to solve differential equations of first order and first degree.

Solve $xdy+ydx \,=\, 0$

$\implies$ $xdy \,=\, -ydx$

$\implies$ $\dfrac{dy}{y} \,=\, -\dfrac{dx}{x}$

$\implies$ $\dfrac{1}{y}dy \,=\, -\dfrac{1}{x}dx$

$\implies$ $\displaystyle \int{\dfrac{1}{y}\,}dy \,=\, -\displaystyle \int{\dfrac{1}{x}\,}dx$

$\implies$ $\log_e{y}+c_1 \,=\, -(\log_e{x}+c_2)$

$\implies$ $\log_e{y}+c_1 \,=\, -\log_e{x}-c_2$

$\implies$ $\log_e{y}+c_1+\log_e{x}+c_2 \,=\, 0$

$\implies$ $\log_e{y}+\log_e{x}+c_1+c_2 \,=\, 0$

$\implies$ $\log_e{(y \times x)}+c \,=\, 0$

$\implies$ $\log_e{(x \times y)}+c \,=\, 0$

$\,\,\, \therefore \,\,\,\,\,\,$ $\log_e{(xy)}+c \,=\, 0$

Latest Math Topics

Nov 03, 2022

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

Oct 24, 2022

Sep 30, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved