Solving ordinary differential equations
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ode::solve(o
,options
) solve(o
,options
)
ode::solve
computes solutions for ordinary
differential equations.
ode::solve(o)
returns the set of solutions
of the ordinary differential equation o
. You can
also call the generic function solve(o)
.
The solver detects the type of the differential equation and
chooses an algorithm according to the detected equation type. If you
know the type of the equation, you can use the option Type
= OdeType
to
pass the equation type to the solver. Passing the equation type to
the solver increases performance.
The solver recognizes the following values of OdeType
:
Abel
 Abel differential equation
Bernoulli
 Bernoulli differential
equation
Chini
 Chini differential equation
Clairaut
 Clairaut differential
equation
ExactFirstOrder
 exact first order
ordinary differential equation
ExactSecondOrder
 exact second
order ordinary differential equation
Homogeneous
 homogeneous first
order ordinary differential equation
Lagrange
 Lagrange differential
equation
Riccati
 Riccati differential
equation
See the Background section for more details on the classes of ordinary differential equations.
If the solver cannot identify the equation with the type you
indicated, it issues a warning and returns the special value FAIL
.
To solve an ordinary differential equation disregarding possible
conditions on the parameters of the equation, use IgnoreSpecialCases
option.
This option eliminates receiving a set of special cases as an answer.
To solve an ordinary differential equation in a simplified manner,
use the IgnoreAnalyticConstraints
option. This
option can provide simple solutions for the equations for which the
direct use of the solver gives complicated results. If you use the IgnoreAnalyticConstraints
option,
always check the answer. This option can lead to wrong or incomplete
results. See Example 3.
The solutions of ordinary differential equations can contain
arbitrary constants of integration. The solver generates the constants
of integration using the format of an uppercase letter C
followed
by an automatically generated number, for example C13
.
The solver does not always verify the uniqueness and completeness of the returned solution. For example:
The solver does not validate the Lipschitzconditions on the ordinary differential equation for the PicardLindelöf Theorem.
For some complex nonlinear systems of differential equations the solver returns constant solutions and does not warn you that other solutions exist.
The solver might ignore assumptions that you set on symbolic
parameters and variables or use them only partially. More precisely, ode::solve
passes
assumptions to the functions that it calls internally. While these
functions can use the specified assumptions, ode::solve
itself
does not use them in most of its internal algorithms. The same happens
if you define an ordinary differential equation using ode
and solve it using solve
.
To define an ordinary differential equation, use the ode
command:
o:= ode(y'(x) = y(x)^2, y(x))
To solve the equation, enter:
ode::solve(o)
or more efficiently:
solve(o)
Internally, the function ode::solve
calls
the function solve
.
You can solve an ordinary differential equation with a symbolic parameter and an initial condition:
o:= ode({y'(x) = a*y(x)^2, y(a) = ln(a)}, y(x)): solve(o)
To reduce the number of returned solutions, use the option IgnoreSpecialCases
.
For example, you can drop the solution for the parameter a =
1:
solve(o, IgnoreSpecialCases)
With the IgnoreSpecialCases
option, a returned
set of solutions can be incomplete.
The solver can return piecewise
solutions:
o:= ode(y'(x) = a/y(x)^2 + b*y(x), y(x)): solve(o)
This solution is complete and mathematically correct for all
possible values of the parameter a and
variable x.
Also you can try the option IgnoreAnalyticConstraints
to
obtain a particular solution that is correct under a set of common
assumptions:
solve(o, IgnoreAnalyticConstraints)
The solver accepts several options:
solve(o, Type = Bernoulli, IgnoreAnalyticConstraints)
Suppose, you want to solve an ordinary differential equation from the class of Bernoulli equations:
o:= ode(y'(x) = ( 1/x + 2*I)*y(x) + 1/x*y(x)^2, y(x)): solve(o)
The solver recognizes the type of the equation and uses the algorithm for solving Bernoulli equations. To improve performance, you can explicitly pass the type of the equation to the solver:
solve(o, Type = Bernoulli)
To solve the Clairaut equation with the initial conditions, enter:
o:= ode({(x*y'(x)y(x))^2  y'(x)^2  1 = 0, y(1) = 1}, y(x)): solve(o, Type = Clairaut)
If the solver cannot identify the equation with the type you
indicated, it issues a warning and returns the special value FAIL
:
o:= ode({(x*y'(x)y(x))^2  y'(x)^2  1 = 0, y(1) = 1}, y(x)): solve(o, Type = Lagrange)
Warning: Cannot detect the Lagrange ODE. [ode::lagrange]
Some ordinary differential equations belong to several classes. For example, some Chini equations are also homogeneous and some Lagrange equations are also Clairaut equations. If an equation belongs to several classes simultaneously, the solver can present its solution in different forms. The form of a solution depends on the class with which an equation is identified. For example, suppose you want to solve the Chini differential equation. You can explicitly pass the type of the equation to the solver:
o:= ode(y'(x) = 1/x*y(x)^2 + 1/x*y(x) + x, y(x)): L:= solve(o, Type = Chini)
You also can let the solver recognize the type of the equation:
solve(o)
The solver does not return the type with which an ordinary differential
equation is internally identified. If you want to verify that both
solution sets are equivalent, use the rewrite
function with target exp
on the first set
of solutions:
rewrite(L, exp)
MuPAD^{®} solves some classes of Riccati ordinary differential
equations that involve arbitrary functions. For example, the following
equation contains the arbitrary function f(x)
:
eq := diff(y(x), x)  f(x)*y(x)^2 + a^2*x^(2*n)*f(x)  a*n*x^(n  1)
For this equation the solver returns:
solve(ode(eq,y(x)))
You also can solve an equation with more than one arbitrary
function. For example, the following equations contain f(x)
and g(x)
:
eq:= diff(y(x), x)  g(x)*f(x)*y(x)  g(x)  diff(f(x), x)*y(x)^2
The returned solution is:
solve(ode(eq,y(x)))
Suppose, you want to solve the following secondorder ordinary differential equation:
eq := x^2*(x^2 + 1)*diff(y(x),x,x) + x*(2*x^2 + 1)*diff(y(x),x)  (nu*(nu + 1)*x^2 + n^2)*y(x)
The solver returns the result in terms of the hypergeometric
function _{1}F_{2} (see hypergeom
):
solve(ode(eq,y(x)))
The solver can handle some third and higherorder ordinary differential equations. For example, solve the following thirdorder linear differential equations:
eq := ode(sin(x)*y'''(x) + cos(x)*y(x), y(x)): solve(eq)
eq := ode(6*y(x) + x^3*y'''(x), y(x)): solve(eq)
The solver also can handle some nonlinear firstorder ordinary differential equations. For example, solve the following firstorder linear differential equations:
eq := ode(y(x)*diff(y(x), x)  y(x)  x^3  4*x^4  4*x^7, y(x)): solve(eq)
eq := ode(exp(x/2)/4  2*exp(x)  y(x) + x*exp(x/2) + y(x)*y'(x), y(x)): solve(eq)

An ordinary differential equation, an object of the type 

Option, specified as Indicates the type of the ordinary differential equation and
accepts the following arguments: 

Option, specified as Pass the option to the generic solver, which is called internally
for all intermediate equations. See the list of options for the 

Pass the option to the generic solver, which is called internally
for all intermediate equations, and to the integrator 

Pass the option to the generic solver, which is called internally
for all intermediate equations, and to the integrator 
Set of solutions of the ordinary differential equation or the
special value FAIL
.
For additional information on the return values, see the solve
help page.
For more information on the particular classes of ordinary differential equations see:
E. Kamke: Differentialgleichungen: Lösungmethoden und Lösungen. B.G. Teubner, Stuttgart, 1997
G.M. Murphy: Ordinary differential equations and their solutions. Van Nostrand, Princeton, 1960
Andrei D. Polyanin and Valentin F. Zaitsev: Handbook of exact solutions for ordinary differential equations, second ed., Chapman & Hall/CRC, Boca Raton, FL, 2003
D. Zwillinger: Handbook of differential equations. San Diego: Academic Press, 1992