How to solve first order linear differential equation

A first order differential equation is an ordinary differential equation (ODE) that involves only the first derivatives of an unknown function with respect to a single independent variable.

Mathematically, first order ODE is represented as:

For example:

Solving first order ODE

Suppose we have the following ODE to solve:

Bring it in the $\frac{dx}{dy} + P(x)y = Q(x)$ form:

Substitute $\frac{dx}{dy} = u\frac{dv}{dx} + v\frac{du}{dx} \textnormal{ and } y=uv$ in the equation above:

Assume $v$ equals zero:

Separate the variables:

Integrating on both sides:

You can put the constant $c$ on either side of the equation.

Change $c$ to $ln(k)$, where $k$ is another constant as $c$ is.

Put $u$ in Eq (1):

Since we assumed $v$ equals zero, the above equation will become:

Integrating on both sides:

where $d$ is a constant. You can put the constant on either side of the equation. Put $u$ and $v$ in $y=uv$:

Since both $d$ and $k$ are constants, we can replace them with a single letter $C$:

Note: Learn about how to solve second order linear differential equations.