Question

Suppose that y(x) is a nonconstant solution of the autonomous equation dy/dx = f(y) and that c is a critical point of the DE. Discuss: Why can’t the graph of y(x) cross the graph of the equilibrium solution y = c? Why can’t f(y) change signs inone of the subregions discussed on page 40? Why can’t y(x)be oscillatory or have a relative extremum (maximumor minimum)?

Short answer

• Within R the function$f\left(y\right)$is either always positive or always negative.
• Within R the solution$y\left(x\right)$can’t be oscillatory or have a relative extremum.

Step-by-step solution

Step 1:Definition of differential equation

Differential Equation is defined as the equation that consists of the derivatives of one or more dependent functions in respect to one or more independent functions.

Step 2:Solution for the function dy/dx = f(y)

$\frac{dy}{dx}=f\left(y\right)$We are assuming f and $f\text{'}$are continuously functions of y. If $f\left(y\right)$changes

signs within R, thenit must be that within R exists $y=c_1$such that $f\left(c_1\right)=0$So we actually have an equilibrium $y=c_1$within R. This contradicts the definition of R as a region between two consecutive equilibrium solutions. So within R the function$f\left(y\right)$is either always positive or always negative.

Suppose that within R the solution$y\left(x\right)$ is oscillatory or has relative extremum. Then $y\text{'}\left(x\right)$must change signs, from the statement of the differential equation, we have. $\frac{dy}{dx}=f\left(y\right)$However we have already shownis either always positive or always negative. So$y\left(x\right)$ must either always be increasing or always decreasing. Within R the solution $y\left(x\right)$can’t be oscillatory or have a relative extremum.

Therefore,

Within R the function$f\left(y\right)$ is either always positive or always negative.

Within R the solution $y\left(x\right)$can’t be oscillatory or have a relative extremum.