Question

Using the autonomous equation (2), discuss how it is possibleto obtain information about the location of points of infectionof a solution curve

Short answer

The points of inflection occur at the y-value where.$f\text{'}\left(x\right)=0$

Step-by-step solution

Step 1:Critical points

The zeros of the function f in$dy/dx=f\left(y\right)$ are of special importance. Wesay that a real number c is a critical point of the autonomous differential equation$dy/dx=f\left(y\right)$ if it is a zero of f—that is$f\left(c\right)=0$, . A critical point is also called an equilibrium point or stationary point.

Step 2:Points of inflection

Assuming the existence of the second derivative, points of inflection of $y\left(x\right)$occur, where.$y"\left(x\right)=0$ So,

$\begin{array}{c}y"\left(x\right)=0\\ 0=\frac{d^{2}y}{d{x}^2}\\ =\frac{d}{dx}\left(\frac{dy}{dx}\right)\end{array}$

Substitute,$\frac{dy}{dx}=f\left(y\right)$

$\begin{array}{c}0=\frac{d}{dx}\left(f\right(y\left)\right)\\ =f\text{'}\left(y\right)\frac{dy}{dx}\\ =f\text{'}\left(y\right)y\text{'}\left(x\right)\end{array}$

Therefore, $f\text{'}\left(y\right)y\text{'}\left(x\right)=0$,

We get

$f\text{'}\left(x\right)=0,y\text{'}\left(x\right)=0$

The solutions to $y\text{'}\left(x\right)=0$are not points of inflection, because they use the first derivative, not the second. So the points of inflection occur at the y-value, where.$f\text{'}\left(x\right)=0$

Hence, the points of inflection occur at the y-value where.$f\text{'}\left(x\right)=0$