Differential Equations and Slope Fields

Starting point: what you already know

• A derivative $\frac{dy}{dx}$ is the gradient of the curve at each point.
• Integration undoes differentiation: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$, $\int \frac{1}{y}dy=\ln |y|+C$, $\int \frac{dy}{1+y^2}={\tan }^{-1}y+C$.
• Exponentials and logs: if $\ln |y|=k$ then $y=\pm e^k$.
• Growth towards a level (previous chapter): $\frac{dN}{dt}=k(N-P)$ gives $N=P+A{e}^{kt}$.

What a differential equation is

A differential equation is an equation with a derivative in it, such as $\frac{dy}{dx}=2x$ or $\frac{dy}{dx}=xy$. Its solution is a function $y$ that makes the equation true.

• The general solution has a constant $C$. It is a whole family of curves.
• A particular solution uses a starting point, such as $y=3$ when $x=1$, to fix $C$.

Slope fields

A slope field draws a short line at many points, each with the gradient the equation gives there. Solution curves follow the little lines, like leaves floating along a stream.

Separating the variables

When the right side is a function of $x$ times a function of $y$, move all the $y$ parts to one side and all the $x$ parts to the other, then integrate both sides.

Quick summary

• A differential equation links a function and its derivative. Its solution is a function.
• General solutions have a constant. A starting point gives the particular solution.
• A slope field shows the gradient at many points. Solution curves follow it.
• Separable: $\int \frac{1}{g(y)}dy=\int f(x)dx$.
• Logistic growth levels off at the carrying capacity $K$ and grows fastest at $\frac{K}{2}$.

Quiz