This section explains how to solve differential equations, called the separable equation.
This is the most fundamental way of solving differential equations, and is a concept that you want to be sure to master, as the goal is to attribute the solution of more complex differential equations to this form by transforming the equation.
Definition
A differential equation that can be written in the form of
$$\frac{dy}{dx}=P(x)Q(y)$$
when \(y\) is a function of \(x\), is called the separable equation.
Solution.
Transform the equation so that \(y\) is on the left-hand side and \(x\) is on the right-hand side, and integrate both sides (\(C\) is an arbitrary constant).
$$\int\frac{1}{Q(y)}dy=\int P(x)dx+C$$
Example
Solve
$$\frac{dy}{dx}=\frac{(x-1)y}{(y+1)x}$$
This is a separable equation that can be written as
$$P(x)=\frac{x-1}{x}, Q(y)=\frac{y}{y+1}$$
When \(y\neq 0\), then we can migrate and integrate both sides, and let the arbitrary constant be \(C_1\),
$$\int\frac{y+1}{y}dy=\int\frac{x-1}{x}dx+C_1$$
$$\int\left(1+\frac{1}{y}\right)dy=\int\left(1-\frac{1}{x}\right)dx+C_1$$
$$y+\ln|y|=x-\ln|x|+C_1$$
$$C_1=\ln|x|+\ln|y|+y-x$$
$$C_1=\ln|xy\exp(y-x)|$$
$$\exp(C_1)=|xy\exp(y-x)|$$
$$\pm\exp(C_1)=xy\exp(y-x)$$
Therefore, let \(C=\pm\exp(C_1)\),
$$xy\exp(y-x)=C$$
is a solution.
Also, \(y=0\) is also a solution, where \(C=0\).
List of solutions for differential equations
For solutions for other types of differential equations, see the following articles.
Comments