In the form

Alternatively,

Eg:
$y^{\prime }=xy$
$y^{\prime }=x^2$
$y^{\prime }-xy=x\to y^{\prime }=x(y+1)$

Not SDE:
$y^{\prime }=x^2+y^2$

1. Constant solutions: $h(y)=0$
   • $y^{\prime }=F(-x,y)$
   • $F(x,a)=0$ for all x
   • $y=a$ is a constant solution
2. Non-constant solutions:

Solution Technique

1. Rewrite as $\frac{1}{g(y)}dy=h(x)dx$ (provided $g(y)\ne 0$)
2. Integrate both sides

Implicit check:

1. Implicitly differentiate both sides
2. Rearrange for $y^{\prime }$
3. Verify that it matches the original ODE
   See also: Implicit Differentiation

Applications of Separable ODEs