An equation that relates a function with its derivatives.

A graphical representation of a differential equation of the form dy/dx = f(x,y), showing small line segments with slopes determined by f(x,y) at grid points.

A numerical method for approximating the solution to a differential equation with a given initial value, using small step sizes.

The use of mathematical concepts and language to describe a physical situation or phenomenon.

The maximum sustainable population size that an environment can support, represented by L in the logistic equation.

The constant of proportionality that affects the rate of population growth.

The rate of change of y with respect to x, or the derivative of y with respect to x.

A differential equation along with a specified initial condition, used to find a particular solution.

A solution that contains arbitrary constants, representing a family of solutions to the differential equation.

A solution obtained from the general solution by assigning specific values to the arbitrary constants, often determined by initial conditions.

1. Choose a grid of points (x, y). 2. At each point, calculate dy/dx = x + y. 3. Draw a short line segment at each point with the calculated slope. 4. Repeat for all points on the grid.

1. $y_{i+1} = y_i + h cdot f(x_i, y_i)$. 2. $y_1 = 0 + 0.1(1 + 0) = 0.1$. 3. $y_2 = 0.1 + 0.1(1.1 + 0.1) = 0.22$. 4. $y(1.2) ≈ 0.22$.

$y_{i+1} = y_i + h cdot f(x_i, y_i)$

$\frac{dP}{dt} = kP(1 - \frac{P}{L})$

$\frac{dy}{dx} = f(x,y)$