Differential Equations and Slope Fields Cheat Sheet

Key Facts

• A first-order differential equation has the form $\frac{dy}{dx} = f(x,y)$ and gives the slope of $y$ at each point $(x,y)$.
• A slope field is made by drawing a short line segment with slope $f(x,y)$ at many points in the coordinate plane.
• A solution curve to $\frac{dy}{dx} = f(x,y)$ must be tangent to the slope field segment at every point it passes through.
• A separable differential equation can be written as $g(y)\,dy = h(x)\,dx$, then solved by integrating $\int g(y)\,dy = \int h(x)\,dx$.
• The general solution of exponential growth or decay $\frac{dy}{dt} = ky$ is $y = Ce^{kt}$, where $C$ is the initial amount when $t = 0$.
• If $k > 0$ in $y = Ce^{kt}$, the model shows growth, and if $k < 0$, the model shows decay.
• The logistic differential equation $\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$ has carrying capacity $L$ and grows fastest when $P = \frac{L}{2}$.
• An initial condition such as $y(0) = 5$ selects one particular solution from a family of solutions.

Vocabulary

Differential Equation

An equation involving a function and one or more of its derivatives, such as $\frac{dy}{dx} = x + y$.

Slope Field

A graph of short line segments showing the slope given by $\frac{dy}{dx} = f(x,y)$ at many points.

Solution Curve

A curve whose tangent slope at every point matches the differential equation.

Separation of Variables

A method for solving some differential equations by placing all $y$ terms with $dy$ and all $x$ terms with $dx$.

Initial Condition

A given value such as $y(a) = b$ that determines a specific solution from the general solution.

Equilibrium Solution

A constant solution where the derivative is zero, such as $\frac{dy}{dt} = 0$.