First-order differential equation

Definition

The term first-order differential equation is used for any differential equation whose order is 1. In other words, it is a differential equation of the form:

$F(x,y,\frac{dy}{dx})=0$

where $F$ is an expression (function) involving three variables. Note that $F$ must make use of $dy/dx$, but it could ignore $x$ or $y$.

For more on general theory and terminology, see differential equation.

Solution strategies

Solution strategies in particular cases

Below are some formats of equations for which general strategies are known. Note that the letter $f$ is no longer used for the solution function but may be used for other functions.:

Equation type	Degree (if polynomial in highest order derivative)	Quick summary of solution strategy
first-order linear differential equation which in simplified form looks like $y^{\prime }+p(x)y=q(x)$	1	Use the integrating factor $e^{H(x)}$ where $H^{\prime }=p$. The general solution is $y=C{e}^{-H(x)}+e^{-H(x)}\int p(x)e^{H(x)}dx$
separable differential equation which is of the form $y^{\prime }=f(x)g(y)$ (any first-order autonomous differential equation is separable, though there are separable differential equations that aren't autonomous)	1	Separate and solve as $\int \frac{dy}{g(y)}=\int f(x)dx$. Also find solutions corresponding to $y=k$ where $g(k)=0$.
Clairaut's equation which is of the form $y=x{y}^{\prime }+f(y^{\prime })$	need not be polynomial; if polynomial, may have any degree	$y=Cx+f(C)$ with $C\in \mathbb{R}$ (all straight lines) and a single other solution explicitly described as the solution to $x+f(dy/dx)=0$, given by $x=-f^{\prime }(p),y=f(p)-p{f}^{\prime }(p)$ as a parametric curve in terms of $p$.