Dsolve function of sympy

Notes

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User Functions

dsolve The function belongs to this user functions Within the scope of .

These functions use from sympy import * Import global namespace . These functions ( Unlike hint Functions), It is only for ordinary users Sympy.

$sympy.solvers.ode.dsolve$

$$\begin{gathered} (\ast eq\ast ,\ast func=None\ast ,hint{=}^{\prime }defaul{t}^{\prime },simplify=True,ics=None,xi=None, \\ \ast eta=None\ast ,\ast x0=0\ast ,\ast n=6\ast ,kwargs) \end{gathered}$$

Can solve any ( Supported by ) Ordinary differential equations and systems of ordinary differential equations .

For simple ordinary differential equations

Usage

The number of equations in the system of equations is 1,

$dsolve(eq,f(x),hint)$ -> Usage method hint In ordinary differential equations eq To solve the problem f(x).

Details

• eq It can be any ordinary differential equation that can be supported . It can be an equation , It can also be an expression , Suppose it equals 0.
• f(x) Its derivatives make up the differential equation . in the majority of cases , We don't need to provide it , It will be calculated automatically （ If it cannot be calculated, an error will be reported ）
• hint It's what you want dsolve Solution method used . Use $classify\_ode(eq,f(x))$ You can get one ODE All the methods that can be used . By default $classify\_ode()$.
• simplify It can be used $odesimp()$ To simplify the . For more information , See its DocString. for example , Turn this option off to disable func Solution or simplification of any constant . It will still integrate with this hint . Be careful , If this option is enabled , The solution may involve more than ODE The order of more arbitrary constants .
• xi and eta Is an infinitesimal function of an ordinary differential equation . They are infinitesimals of Lie groups of invariant point transformations of differential equations . The user can specify a value for infinitesimal . If not specified ,xi and eta Will be used with the help of various Heuristics $infinitesimals()$ Calculate .
• ics Is the initial of the differential equation / Set of boundary conditions . It should be in a form similar to $\{f(x0):x1,f(x).diff(x).subs(x,x2):x3\}$ give . For power series solutions , If no initial condition is specified , Then assume f(0) by c0, The solution of the power series is 0.
• x0 It's the point that needs to be solved .
• n Give the index of the variable to be solved

Hints

Jump first

Tips

• You can declare the derivative of an unknown function like this :

from sympy import Function,Derivative
from sympy.abc import x    # x Is an independent variable 
f= Function("f")(x)   #f yes x Function of 
f_=Derivative(f,x)   #f_ yes f Yes x The derivative of 

• stay test_ord .py You can see a series of how to use $dsolve()$ An example of .
• $dsolve()$ Always return an equality class ( Except the hint is all or all_Integral The situation of ). If possible , It will explicitly solve for the function being solved . otherwise , It returns an implicit solution . Any constant is called C1、C2 Symbols for, etc .
• Because all solutions are mathematically equivalent , So for ODE, Some prompts may return identical results . however , Usually two different prompts return the same solution in different formats . The two should be equivalent . Also pay attention to , Sometimes the values of any constant in two different solutions may be different , Because a constant can be “ absorption ” Other constants .
• use $help(ode.od{e}_{<}hintname>)$ You can get more information about specific tips , One of them is without _Integral The name of the prompt for .

For ordinary differential equations

Usage

$dsolve(eq,func)$ -> Xie Changyou x(t),y(t),z(t) And so on eq.

Details

eq It can be an arbitrarily supported system of ordinary differential equations. It can be an equation , It can also be an expression , Suppose it equals 0.

func set up x(t) and y(t) Is a function of a variable , The variable and its partial derivatives constitute a system of ordinary differential equations . It will be automatically detected ( If you cannot detect , Will cause an error ).

Examples

from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols
from sympy.abc import x
f=Function('f')
ans=dsolve(Derivative(f(x),x,x)+9*f(x),f(x))
print(ans)
#Eq(f(x), C1*sin(3*x) + C2*cos(3*x))