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# 21. Differential Equations

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## 21.1 Introduction to Differential Equations

This section describes the functions available in Maxima to obtain analytic solutions for some specific types of first and second-order equations. To obtain a numerical solution for a system of differential equations, see the additional package `dynamics`. For graphical representations in phase space, see the additional package `plotdf`.

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## 21.2 Functions and Variables for Differential Equations

Function: bc2 (solution, xval1, yval1, xval2, yval2)

  Solves a boundary value problem for a second order differential equation. Here: solution is a general solution to the equation, as found by `ode2`; xval1 specifies the value of the independent variable in a first point, in the form `x = x1`, and yval1 gives the value of the dependent variable in that point, in the form `y = y1`. The expressions xval2 and yval2 give the values for these variables at a second point, using the same form.

  See `ode2` for an example of its usage.

  ·

  @ref{Category: Differential equations}

Function: desolve (eqn, x)
Function: desolve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])

  The function `desolve` solves systems of linear ordinary differential equations using Laplace transform. Here the eqn's are differential equations in the dependent variables x_1, ..., x_n. The functional dependence of x_1, ..., x_n on an independent variable, for instance x, must be explicitly indicated in the variables and its derivatives. For example, this would not be the correct way to define two equations:

      eqn_1: 'diff(f,x,2) = sin(x) + 'diff(g,x);
      eqn_2: 'diff(f,x) + x^2 - f = 2*'diff(g,x,2);

  The correct way would be:

      eqn_1: 'diff(f(x),x,2) = sin(x) + 'diff(g(x),x);
      eqn_2: 'diff(f(x),x) + x^2 - f(x) = 2*'diff(g(x),x,2);

  The call to the function `desolve` would then be

      desolve([eqn_1, eqn_2], [f(x),g(x)]);

  If initial conditions at `x=0` are known, they can be supplied before calling `desolve` by using `atvalue`.

      (%i1) 'diff(f(x),x)='diff(g(x),x)+sin(x);
                       d           d
      (%o1)            -- (f(x)) = -- (g(x)) + sin(x)
                       dx          dx
      (%i2) 'diff(g(x),x,2)='diff(f(x),x)-cos(x);
                        2
                       d            d
      (%o2)            --- (g(x)) = -- (f(x)) - cos(x)
                         2          dx
                       dx
      (%i3) atvalue('diff(g(x),x),x=0,a);
      (%o3)                           a
      (%i4) atvalue(f(x),x=0,1);
      (%o4)                           1
      (%i5) desolve([%o1,%o2],[f(x),g(x)]);
                        x
      (%o5) [f(x) = a %e  - a + 1, g(x) =

                                                      x
                                         cos(x) + a %e  - a + g(0) - 1]
      (%i6) [%o1,%o2],%o5,diff;
                   x       x      x                x
      (%o6)   [a %e  = a %e , a %e  - cos(x) = a %e  - cos(x)]

  If `desolve` cannot obtain a solution, it returns `false`.

  ·

  @ref{Category: Differential equations} · @ref{Category: Laplace transform}

Function: ic1 (solution, xval, yval)

  Solves initial value problems for first order differential equations. Here solution is a general solution to the equation, as found by `ode2`, xval gives an initial value for the independent variable in the form `x = x0`, and yval gives the initial value for the dependent variable in the form `y = y0`.

  See `ode2` for an example of its usage.

  ·

  @ref{Category: Differential equations}

Function: ic2 (solution, xval, yval, dval)

  Solves initial value problems for second-order differential equations. Here solution is a general solution to the equation, as found by `ode2`, xval gives the initial value for the independent variable in the form `x = x0`, yval gives the initial value of the dependent variable in the form `y = y0`, and dval gives the initial value for the first derivative of the dependent variable with respect to independent variable, in the form `diff(y,x) = dy0` (`diff` does not have to be quoted).

  See `ode2` for an example of its usage.

  ·

  @ref{Category: Differential equations}

Function: ode2 (eqn, dvar, ivar)

  The function `ode2` solves an ordinary differential equation (ODE) of first or second order. It takes three arguments: an ODE given by eqn, the dependent variable dvar, and the independent variable ivar. When successful, it returns either an explicit or implicit solution for the dependent variable. `%c` is used to represent the integration constant in the case of first-order equations, and `%k1` and `%k2` the constants for second-order equations. The dependence of the dependent variable on the independent variable does not have to be written explicitly, as in the case of `desolve`, but the independent variable must always be given as the third argument.

  If `ode2` cannot obtain a solution for whatever reason, it returns `false`, after perhaps printing out an error message. The methods implemented for first order equations in the order in which they are tested are: linear, separable, exact - perhaps requiring an integrating factor, homogeneous, Bernoulli's equation, and a generalized homogeneous method. The types of second-order equations which can be solved are: constant coefficients, exact, linear homogeneous with non-constant coefficients which can be transformed to constant coefficients, the Euler or equi-dimensional equation, equations solvable by the method of variation of parameters, and equations which are free of either the independent or of the dependent variable so that they can be reduced to two first order linear equations to be solved sequentially.

  In the course of solving ODE's, several variables are set purely for informational purposes: `method` denotes the method of solution used (e.g., `linear`), `intfactor` denotes any integrating factor used, `odeindex` denotes the index for Bernoulli's method or for the generalized homogeneous method, and `yp` denotes the particular solution for the variation of parameters technique.

  In order to solve initial value problems (IVP) functions `ic1` and `ic2` are available for first and second order equations, and to solve second-order boundary value problems (BVP) the function `bc2` can be used.

  Example:

      (%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;
                            2 dy           sin(x)
      (%o1)                x  -- + 3 x y = ------
                              dx             x
      (%i2) ode2(%,y,x);
                                   %c - cos(x)
      (%o2)                    y = -----------
                                        3
                                       x
      (%i3) ic1(%o2,x=%pi,y=0);
                                    cos(x) + 1
      (%o3)                   y = - ----------
                                         3
                                        x
      (%i4) 'diff(y,x,2) + y*'diff(y,x)^3 = 0;
                               2
                              d y      dy 3
      (%o4)                   --- + y (--)  = 0
                                2      dx
                              dx
      (%i5) ode2(%,y,x);
                            3
                           y  + 6 %k1 y
      (%o5)                ------------ = x + %k2
                                6
      (%i6) ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));
                                   3
                                2 y  - 3 y
      (%o6)                   - ---------- = x
                                    6
      (%i7) bc2(%o5,x=0,y=1,x=1,y=3);
                               3
                              y  - 10 y       3
      (%o7)                   --------- = x - -
                                  6           2

  ·

  @ref{Category: Differential equations}

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