Solution to DEs specifically for first order

Direction Fields

Graphs that give us all possible solutions to a DE

Each line is a possible solution to the DE

And the initial conditios will decide which one we want to use

A constant solution $y=a$ is called an equilibrium solution

A equilibrium solution can be unstable or asymptotically stable depending on the behavior of the solutions next to them

• asymptotically stable = all nearby solutions approach $a$
• unstable = all nearby solutions move away from $a$

If every solution is constant, they're simply all called stable

Autonomous DEs

Substitution

Used in DEs like:

• rational and every term has a in/dependent variable

Substituting $y(x)=v(x)x$
$\frac{dy}{dx}=v^{\prime }x+v$
Then

We then isolate $v^{\prime }$ and can integrate to find the value of v

Then convert back to terms of y