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

$y = F(x) + C$

Solve $\frac{dy}{dx} = 0$ or where $\frac{dy}{dx}$ is undefined.

$\int \frac{1}{1+x^2} dx = \arctan(x) + C$

$y = f(x) + C$, where C is an arbitrary constant.

Use initial condition $y(x_0) = y_0$ to solve for C in $y = f(x) + C$.

Slope at any point (x,y) is the difference between x and y.

$y = \int f(x) dx + C$

Look for horizontal or vertical tangents on the graph.

$\frac{dy}{dx} |_{(x_0,y_0)}$

A visual representation of solutions to differential equations, showing slopes at different points.

A point where the derivative of a function is zero or undefined.

An equation that relates a function with its derivatives.

An arbitrary constant (+C) added during integration, representing a family of solutions.

A set of solutions to a differential equation, each differing by the constant of integration.

Indicate a slope of zero, potential critical points.

Indicate an undefined slope, potential critical points.

Represents the magnitude of the slope.

A specific value used to determine the constant of integration (+C) and find a particular solution.

A single solution from the family of functions, determined by an initial condition.

Slope fields provide a graphical representation of the slope at various points, allowing us to approximate solution curves without explicitly solving the differential equation.

The constant of integration accounts for the fact that the derivative of a constant is zero, leading to a family of possible solutions differing by a constant value.

Initial conditions provide a specific point on the solution curve, allowing us to solve for the constant of integration and identify a unique solution.

The derivative is zero, indicating a potential maximum or minimum.

Positive slopes indicate increasing function, negative slopes indicate decreasing function.

Denser segments indicate faster changes in the function's value.

Slope field visually represents the derivative of a function at various points.

Observe how the slopes change; increasing slopes indicate concave up, decreasing slopes indicate concave down.

Equilibrium solutions are constant solutions where $\frac{dy}{dx} = 0$ for all x, represented by horizontal lines in the slope field.

It represents the vertical shift of the solution curve.