Autonomous Differential Equations

Differential equations of the form are called autonomous.
or an autonomous equation, the solutions y(t) = yi, where yi’s are the
roots of f (t), are called equilibrium.

  • If an equilibrium repels neighboring solutions as t → ∞ then it is called a source, repeller, or unstable equilibrium,
  • If an equilibrium attracts neighboring solutions as t → ∞ then it is called a sink, attractor, or stable equilibrium,
  • If an equilibrium is neither a source nor a sink, then it is called a node, or semi-stable.

Equilibrium solution

Occurs at the points where

Chapter 2: First-Order ODEs

Three main types of First-Order Equations

Separable ODE

Linear (1st order) ODE

Exact ODE

where

Separable ODEs



Assume that
is inseparable, so this is NOT a separable ODE

Solving Separable ODEs

“Separate then integrate”

  1. Convert the given equation into the form
  2. Integrate both sides to obtain . This gives the implicit solution to the given equation. (General solution in implicit form)
  3. Convert the solution obtained in step 2 to explicit form and use the initial condition to solve for the constant C, if required/able.

Solve the following IVP:

  1. Separate
  2. Integrate
  3. Use initial condition to solve for C:
    Thus, the particular soln in implicit form is
    To convert this into explicit form:




    There can only be 0, 1, or infinitely many solutions. Not 2, so we have to figure out if we are using the + or - sign.
    Under the inital condition when , which is if the sign is +, or if the sign is -
    So the correct sign is -
    is the particular solution in explicit form.

  1. Separate
  2. Integrate