Interval of validity

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Intros
Lessons
  1. What is an interval of validity?
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Examples
Lessons
  1. Determining Intervals of Validity
    For each of the following differential equations, the corresponding slope field is provided.
    Sketch the solution for each differential equation with the specific initial conditions given. What is the interval of validity?
    1. dydx=1(x+1),y(0)=2 \frac{dy}{dx}=\frac{1}{(x+1)}, y(0)=2
      Determining the interval of validity with slope fields
    2. dydx=e2x2+2y,y(4)=4 \frac{dy}{dx}=e^{-2x^2}+2y, y(-4)=4
      Slope fields and interval of validity
  2. In the section to do with Bernoulli Equations the solution to the differential equation:

    dydx+1xy=y3\frac{dy}{dx}+\frac{1}{x} y=y^3

    with initial condition y(12)=1y(\frac{1}{2})=1 was found to be:

    y=12xy=\frac{1}{\sqrt{2x}}

    Plot out this solution and explicitly give its interval of validity.
    1. Solve the following differential equation:

      dydx=2xey\frac{dy}{dx}=-2xe^{-y}

      With initial conditions y(1)=0y(1)=0.

      What is the interval of validity for the solution?
      1. Solve the following differential equation:

        dydx=4xy2\frac{dy}{dx}=4xy^2
        1. What is the interval of validity with initial conditions y(0)=12y(0)=\frac{1}{2}?
        2. What is the interval of validity with initial conditions y(12)=2y(\frac{1}{2})=-2?