Separable Equation [Solution of differential equation Vol. 1]

How to Solve Differential Equation

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.


A differential equation that can be written in the form of


when \(y\) is a function of \(x\), is called the separable equation.


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).

$$\int\frac{1}{Q(y)}dy=\int P(x)dx+C$$




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 \(C_1\),








Therefore, let \(C=\pm\exp(C_1)\),


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.