Notebook

# Maths

## Multivariable Calculus and ODEs

The following chapter shows Ordinary Differential Equations. The code used can be found here.

In [1]:
% run ./maths_functions.py


### First Order Differential Equations (ODEs)

A F.ODE is generalized like this:

$a.y'+ b.y = c$

Steps to solve:

1) Solve for $c=0$, $y=1$, $y'=m$

2) Get $y_1 = A.e^{mx}$

3) Substitute left according to right, eg. for $y=2.sin(3x)$, substitute $y_2=B.sin(\alpha x) + C.cos(\alpha x)$

4) Find $\alpha$ and compute $y_2$

5) General Solution is $y_{GS}=y_1+y_2$

6) Particular solution is dependent on conditions.

### Second Order Differential Equations

Steps to solve:

1) Similar to F.ODE, but substitute $y''=m^2$ and get two roots.

2) if different real m-roots, $y=P.e^{m_1.t}+Q.e^{m_2.t}$.

3) if ($m_1==m_2$), $y=(P.t+Q).e^{mt}$

4) if complex roots, $y= e^{\lambda t}.(P.cos(\mu t) + Q.sin(\mu t))$, where $m=\lambda \pm \mu.j$

5) Solve like F.ODE 3)-6)