How

How to create a symbolic function

To create a symbolic function use sympy.Function.

Tip

sympy.Function("y")

For example:

import sympy as sym

y = sym.Function("y")
y

y

We can pass symbolic variables to this symbolic function:

x = sym.Symbol("x")
y(x)

\[\displaystyle y{\left(x \right)}\]

We can create the derivative of a symbolic function:

sym.diff(y(x), x)

\[\displaystyle \frac{d}{d x} y{\left(x \right)}\]

How to create a differential equation

To create a differential equation we use sympy.Eq.

Tip

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: \(\frac{dy}{dx} = \cos(x) y\) we would 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

\[\displaystyle \frac{d}{d x} y{\left(x \right)} = y{\left(x \right)} \cos{\left(x \right)}\]

How to obtain the general solution of a differential equation

To obtain the generation solution to a differential equation we use: sympy.dsolve.

Tip

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: \(\frac{dy}{dx} = \cos(x) y\) we would 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))

\[\displaystyle y{\left(x \right)} = C_{1} e^{\sin{\left(x \right)}}\]

How to obtain the particular solution of a differential equation

To obtain the particular solution to a differential equation we use: sympy.dsolve and pass the initial conditions: ics.

Tip

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 \(y(5)= \pi\) we would 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)

\[\displaystyle y{\left(x \right)} = \pi e^{- \sin{\left(5 \right)}} e^{\sin{\left(x \right)}}\]

Note

The syntax used here is similar to the way we substitute values in to algebraic expressions.