### Differential equations: Direction field

### Autonomous ODEs

We have seen that a first-order ODE of degree one can be written as \[y'= \varphi(t,y)\] for a two-variable \(\varphi\) function (if we restrict the domain of the unknown function #y# in such a way that the coefficient #f(t,y)# of #y'# in the general form #f(t,y)\cdot y'=g(t,y)# of the ODE has no zeroes). If #\varphi# does not depend on the second argument, #y#, then solving the equation amounts to computing the antiderivative of the function \(\varphi\) of #t#. The other extreme is the following case.

Autonomous, stable, unstable, and semi-stable

A first-order ODE is called **autonomous** if it can be written as \[ y'=\varphi(y)\] for a function #\varphi#. In other words, if it is a first-order first-order ODE that does not depend on the independent variable.

Constant functions that are solutions of the equation #\varphi(y)=0#, are called **equilibrium solutions**. The corresponding solution curves are horizontal lines in the direction field of the ODE.

- If all solutions in the vicinity of an equilibrium solution converge to the equilibrium, then the solution is called
**stable**. - If all solutions close to equilibrium equilibrium run away from the equilibrium (diverge), then the solution is called
**unstable**. - If all solutions on one side of the equilibrium diverge from that equilibrium and all solutions on the other side of the equilibrium converge to the equilibrium, then the solution is called
**semi-stable**.

If #\varphi# is continuous and #y=a# is the only equilibrium solution in an open interval around #a#, then this equilibrium solution is stable, unstable, or semi-stable.

It may happen that an equilibrium solution occurs at the boundary of an interval of values of #y# for which solutions occur. The notion of semi-stability will then coincide with stability or instability, as there are no solutions on one side of the equilibrium. An example can be found at the bottom of this page.

The derivative of any solution of an autonomous ODE is not dependent on the independent variable (think of the time #t#). This means that the direction field has the first property described below.

Properties of Autonomous ODEs

The direction field of a first order autonomous differential equation #y'=\varphi(y)# does not change under horizontal shifts.

If #y# is a solution of this ODE and #c# is a constant, then the function #y_c# defined by #y_c(t)=y(t-c)# is also a solution.

The nature of an equilibrium solution #y=a# is determined by the sign of #\varphi(a+\delta)# for values of #\delta# close to #0#:

- If #\varphi(a+\delta)\lt0# for all positive #\delta# close to #0# and #\varphi(a+\delta)\gt0# for all negative #\delta# close to #0#, then #y=a# is stable.
- If #\varphi(a+\delta)\gt0# for all positive #\delta# close to #0# and #\varphi(a+\delta)\lt0# for all negative #\delta# close to #0#, then #y=a# is unstable.
- If #\varphi(a+\delta)# has the same sign for all #\delta\ne0# close to #0#, then #y=a# is semi-stable.

Can you find the specific solution with initial value \(y(0)=1\)?

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