Slope fields

Slope fields

Lessons

Slope fields, also called directional fields or vector fields, are graphical representations of first-order differential equations.

Slope Fields consist of a bunch of lines indicating the slope of y with respect to x, or dydx\frac{dy}{dx}
  • 1.
    What are Slope Fields?

  • 2.
    Understanding Slope Fields
    Find the directional field for the following equations:
    a)
    dydx=xyx \frac{dy}{dx}=xy-x

    b)
    dydx=x2(y+1) \frac{dy}{dx}=\frac{x^2}{(y+1)}


  • 3.
    Determining the Equation from a Slope Field
    Which equation best corresponds to the following slope field?
    Slope fields and corresponding differential equations

    i.
    dydx=y2\frac{dy}{dx}=y-2
    ii.
    dydx=xy2\frac{dy}{dx}=xy-2
    iii.
    dydx=x+1\frac{dy}{dx}=x+1
    iv.
    dydx=x+1\frac{dy}{dx}=-x+1

  • 4.
    Which equation best corresponds to the following slope field?
    determine the best corresponds differential equations of slope fields

    i.
    dydx=xy2\frac{dy}{dx}=xy-2
    ii.
    dydx=yx\frac{dy}{dx}=-\frac{y}{x}
    iii.
    dydx=2x+y\frac{dy}{dx}=2x+y
    iv.
    dydx=xy3\frac{dy}{dx}=xy-3

  • 5.
    Given the differential equation and its resulting slope field:
    dydx=y2(y3)\frac{dy}{dx}=\frac{y}{2}(y-3)
    Slope fields and differential equations

    Draw a solution to the following differential equation using the following initial value conditions:
    a)
    y(2)=1y(-2)=1

    b)
    y(0)=4 y(0)=4

    c)
    y(1)=3 y(1)=3