Differential Equations

Differential equations week

First meeting

After this meeting students should:

• Understand how to write a differential equation.
• Understand how to use the sympy library to solve differential equations to obtain both the general and particular solution.

Problem

Explain to students that we will be solving the following problem:

Obtain the general solution to the following differential equation:

$$\frac{dy}{dx}=3y^2-5xy^2$$

Obtain the particular solution given that $y(3)=4$.

Set up the variable and the function:

>>> import sympy as sym
>>> x = sym.Symbol("x")
>>> y = sym.Function("y")

Write the differential equation:

>>> differential_equation = sym.Eq(lhs=sym.diff(y(x), x), rhs=3 * y(x) ** 2 - 5 * x * y(x) ** 2)
>>> differential_equation
Eq(Derivative(y(x), x), -5*x*y(x)**2 + 3*y(x)**2)

Use sympy.dsolve to obtain the general equation:

>>> sym.dsolve(differential_equation, y(x))
Eq(y(x), 2/(C1 + 5*x**2 - 6*x))

Create the boundary conditions:

>>> condition = {y(3): 4}
>>> sym.dsolve(differential_equation, y(x), ics=condition)
Eq(y(x), 2/(5*x**2 - 6*x - 53/2))

Come back: with time take any questions.

Point at resources.