Differential Equations

The function's behavior at specific points.

The function's symmetry.

The function's domain.

The function's range.

Separation of variables to find an explicit solution.

Estimating using nearby slopes in the slope field.

Using Euler's Method with a small step size starting from $(x_0, y_0)$.

Sketching isoclines for several values close to $y_0$.

y = 2\sin(x) + 3

y = 2\cos(x) + 3

y = 2\sin(x) + 2

y = 2\cos(x) + 2

The rate of change described by its associated differential equation is constant for all values of x.

There exists an asymptote for every solution curve to this differential equation along those diagonal lines.

The solution curves for this differential get steeper as x increases across these diagonal lines.

The solution curves have inflection points wherever they intersect these diagonal lines.

Solution curves will eventually diverge as x increases without bound.

There are possible discontinuities in solution curves further out along x-axis values as they become more parallel over time.

The differential equation likely has horizontal asymptotes as x goes towards infinity.

Solution curves will eventually converge into one single curve as x increases without bound.

The slopes vary with changes in x but not y within the given differential equation.

The corresponding differential equation has constant nonzero slope regardless of $x$ or $y$ value.

Each point on any curve represented by this field will have an infinite number of tangents.

There is no particular solution to the corresponding differential equation.

A fluid flowing in a specific direction

The curvature of a graph

The stability of a system

The rate of change at different coordinate points

To perform algebraic manipulations

To graphically visualize the slope field

To determine the initial conditions of the system

To obtain precise numerical solutions

y = 2x^2 - 6x + 7

y = 2x^2 - 6x + 1

y = 2x^2 - 6x + 3

y = 2x^2 - 6x + 5

Solution curves form circles around non-equilibrium points suggesting cyclical behavior

Solved curves move away from equilibrium points indicating unstable equilibria

Solution curves remain straight regardless of nearby equilibrium points

Solution curves converge onto equilibrium points indicating stable equilibria