First-order Ordinary Differential Equations

First-order ODEs

A first-order ODE has the form

$$F(x,y,y^{\prime })=0$$

Solving it usually produces a one-parameter family of curves

$$\varphi (x,y,C)=0\text{or}y=y(x;C)$$

where $C$ is the constant of integration

Each value of $C$ gives one solution curve.

Geometrically, the ODE defines a slope field and the family $\varphi (x,y,C)=0$ is a collection of curves tangent to that field.