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MATLAB Tutor

Part 12: Symbolic solution of differential equations

In this part we explore MATLAB's ability to solve the logistic equation

dy/dt = y (1 - y)

and to check the solution. Then we will adapt the solution procedure to an initial value problem with this same differential equation. In the next part, we will relate these algebraic calculations to the geometry of direction fields.

  1. First declare the variable t as symbolic and give the differential equation a name by entering:

    syms t
    DE1 = ' Dy = y*(1 - y) '


    Then ask MATLAB to solve the differential equation by entering:

    ysol = dsolve(DE1)
    pretty(ysol)     % Gives nice format


    Note that MATLAB uses C1 to represent an arbitrary constant.


  2. Differentiate your solution expression with respect to t to get an explicit expression for dy/dt. You can can differentiate ysol and simplify the result by using:

    dysol = simple( diff(ysol) )
    pretty(dysol)     % Gives nice format


    Then use your solution expression ysol to find an explicit formula in t for y(1 - y). Is this formula the same as the one for dy/dt? You may want to simplify the output before you try to answer this.


  3. Now we add the initial condition y(0) = 1/10 to determine a single solution of the differential equation. To tell MATLAB to solve the initial value problem, put both parts of the problem -- the differential equation and the initial condition -- into dsolve. Enter:

    h = dsolve( DE1, ' y(0) = 1/10 ' )
    pretty(h)     % Gives nice format


  4. What do you have to do to check the answer from the preceding step? Have you done it already? If not, can you get the checking technique from what you did in Step 2?

    To substitute t = 0 into the solution function h, enter:

    subs(h, t, 0)

    We will use our solution function h for further computations in Part 13, which follows next.

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