How to
Contents
How to#
Create a symbolic function#
To create a symbolic function use sympy.Function
.
Usage
sympy.Function("y")
For example:
import sympy as sym
y = sym.Function("y")
y
y
You can pass symbolic variables to this symbolic function:
x = sym.Symbol("x")
y(x)
You can create the derivative of a symbolic function:
sym.diff(y(x), x)
Create a differential equation#
To create a differential equation use sympy.Eq
.
Usage
import sympy as sym
y = sym.Function("y")
x = sym.Symbol("x")
equation = sym.Eq(lhs, rhs)
Where lhs
and rhs
are expressions in \(y\), \(\frac{dy}{dx}\) and \(x\).
For example to create the differential equation: \(\frac{dy}{dx} = \cos(x) y\) write:
import sympy as sym
y = sym.Function("y")
x = sym.Symbol("x")
lhs = sym.diff(y(x), x)
rhs = sym.cos(x) * y(x)
differential_equation = sym.Eq(lhs, rhs)
differential_equation
How to obtain the general solution of a differential equation#
To obtain the generation solution to a differential equation use:
sympy.dsolve
.
Usage
import sympy as sym
y = sym.Function("y")
x = sym.Symbol("x")
equation = sym.Eq(lhs, rhs)
sym.dsolve(equation, y(x))
For example to solve the differential equation: \(\frac{dy}{dx} = \cos(x) y\) write:
import sympy as sym
y = sym.Function("y")
x = sym.Symbol("x")
lhs = sym.diff(y(x), x)
rhs = sym.cos(x) * y(x)
differential_equation = sym.Eq(lhs, rhs)
sym.dsolve(differential_equation, y(x))
Obtain the particular solution of a differential equation#
To obtain the particular solution to a differential equation use:
sympy.dsolve
and pass the initial conditions: ics
.
Usage
import sympy as sym
y = sym.Function("y")
x = sym.Symbol("x")
equation = sym.Eq(lhs, rhs)
sym.dsolve(equation, y(x), ics={y(x_0): value})
For example, to solve the differential equation: \(\frac{dy}{dx} = \cos(x) y\) with the condition \(y(5)= \pi\) write:
import sympy as sym
y = sym.Function("y")
x = sym.Symbol("x")
lhs = sym.diff(y(x), x)
rhs = sym.cos(x) * y(x)
differential_equation = sym.Eq(lhs, rhs)
condition = {y(5): sym.pi}
sym.dsolve(differential_equation, y(x), ics=condition)
Note
The syntax used here is similar to substituting values in to algebraic expressions.