Slope fields: notes and practice questions

A slope field for the differential equation $\frac{dy}{dx} = y - x^2$ is shown.

Some of the solutions to the differential equation have a local maximum point and a local minimum point.

Write down the equation of the curve on which all these maximum and minimum points lie.

Sketch this curve on the slope field.

The solution to the differential equation that passes through the point (0, 1) has both a local maximum point and a local minimum point.

On the slope field, sketch the solution to the differential equation that passes through (0, 1).

A biologist is modeling the population growth rate of two interacting species, $X$ and $Y$, in a confined ecosystem. The rate of change of species $Y$ with respect to species $X$ is described by the differential equation $\frac{dy}{dx} = \frac{x^2 - y^2}{2xy}$, where $x$ and $y$ represent the populations of species $X$ and $Y$ respectively, and $x > 0, y > 0$.

In a slope field, an isocline is defined as a curve on which the gradient is constant. In other words, an isocline is a curve where $\frac{dy}{dx} = k$, with $k$ a constant.

(a) State which of the following equations are isoclines of the above differential equation.

(i) $y = x$

(ii) $y = -x$

(iii) $y = 2x$

(iv) $x^2 - y^2 - 4xy = 0$

(v) $x = 1$

(b) Determine whether the solution to the differential equation $\frac{dy}{dx} = \frac{x^2 - y^2}{2xy}$, $x > 0, y > 0$ has any turning points. Justify your answer.

A biologist is studying the growth of a new bacterial colony in a controlled environment. The rate of change of the population $P$ with respect to time $t$ is modelled by the differential equation $\frac{dP}{dt} = \frac{2t}{e^P}$, where $P$ is the population size (in thousands) and $t$ is the time in hours.

(a) Identify which of the following diagrams, A, B or C, represents the slope field for the differential equation. Give a reason for your answer.

(b) It is given that, at time $t=0$ hours, the initial population is $P=0$ (in thousands). Find an expression for $P$, in terms of $t$, for this solution.

(c) Find $\frac{dP}{dt}$, in terms of $t$, by differentiating your answer from part (b).

(d) Hence verify that your answer to part (b) is a solution to $\frac{dP}{dt} = \frac{2t}{e^P}$.