What is the general form of a first-order ordinary differential equation?

Key concept: A first-order differential equation is one of the first places where notation matters. Once the roles of the variables are clear, the definition becomes straightforward.

Step by step

A first-order ordinary differential equation (ODE) is an equation involving:

• an unknown function, usually written as $y(x)$,
• its first derivative $y'(x)$,
• and often the independent variable $x$ itself.

A common general form is

$y'(x)=F(x,y).$

Here:

• $x$ is the independent variable,
• $y$ is the dependent variable, meaning $y=y(x)$,
• $F(x,y)$ is some function of the two variables.

Sometimes the same idea is written as

$\frac{dy}{dx}=F(x,y).$

Both forms mean the same thing.

If the equation can be written as $y'=f(x)$ only, then it is still first-order, just a simpler special case.

Pitfall alert

It is easy to think $F$ must be a function of only $x$ or only $y$, but in the general case it can depend on both.

Also, the order is determined by the highest derivative present. If only $y'$ appears, it is first-order; if $y''$ appears, it is second-order.

Try different conditions

If the equation is written implicitly, such as

$M(x,y)\,dx+N(x,y)\,dy=0,$

it may still be a first-order ODE after dividing by $dx$ when possible:

$N(x,y)\frac{dy}{dx}+M(x,y)=0.$

So the same first-order idea can appear in several notations.

Further reading

ordinary differential equation, dependent variable, first derivative