Example: How to Solve an Initial Value Problem (Exponential Growth)

  1. Find the solution of the differential equation dy/dt=2y that satisfies y(0)=1.
  2. Sketch the graph of the solution along with the slope field for dy/dt=2y.
  3. Terminology:
    1. What do we call a problem such as dy/dt=2y, y(0)=3?
    2. What do we call the equation y(0)=3?
  4. Solve dy/dt=(1/3)y, y(0)=2 (Use a formula to find the solution quickly.)

Answer:

  1. The solution is y=e^(2t). In the video, a full explanation is given for how this solution is derived. Near the end of the video, you'll learn a formula based on this derivation that will enable you to quickly solve the important differential equation dy/dt=ry for any constant r.

  2. The graph of the solution, along with the slope field.

    Graph of the Solution y=e^(2t) to the Differential Equation dy/dt=2y Satisfying y(0)=1, with Slope Field
  3. Terminology:
    1. We call this an initial value problem.
    2. We call this the initial condition.
  4. The solution is y=2e^((1/3)t).

Icon for Higher Math Notes: Differential Equations (A Higher Math Help Resource)

Video Explanation

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