Next: Functions and Variables for Differential Equations, Previous: Differential Equations, Up: Differential Equations [Contents][Index]
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.
Previous: Introduction to Differential Equations, Up: Differential Equations [Contents][Index]
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.
The function desolve solves systems of linear ordinary
differential equations using Laplace transform. Here the eqn’s are
differential equations in the dependent variables y_1, ...,
y_n. The functional dependence of y_1, ..., y_n on an
independent variable, for instance x, must be explicitly indicated
in the variables and its derivatives. For example, the correct
way to define the differential equations 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.
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.
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.
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.
See also desolve, drawdf and rk.
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
The variable method is set by ode2 to the successful solution
method.
%c is the integration constant in the solutions of first
order ODEs returned from ode2.
%k1 is the first integration constant in the solutions of second
order ODEs returned from ode2.
%k2 is the second integration constant in the solutions of second
order ODEs returned from ode2.
yp is the particular solution of an ODE found by ode2
when using the variation of parameters technique.