Takes the form:

$$\frac{d^2y}{dt^2} + a(t)\frac{dy}{dt} + b(t)y = c(t)$$

With initial conditions $y(0) = y_0$ and $y^\prime(0) = y^\prime_0$.

Coefficients are functions of $t$ and are defined in second_order.ipynb

second_order_ode module in the solve.py file in the ode directory, and is imported into second_order.ipynb.

Takes the form:

$$\frac{dx}{dt} = F(x, y)$$

$$\frac{dy}{dt} = G(x, y)$$

With initial conditions $x(0) = x_0$ and $y(0) = y_0$.

$F$ and $G$ are defined in first_order_system.ipynb.

first_order_system_2vars module in the solve.py file in the ode directory, and is imported into first_order_system.ipynb.

Then for 3 variable systems, use the module first_order_system_3vars.

1. numpy

2. scipy using the integrate module

3. scipy using the optimize module

4. matplotlib using the pyplot module