Added and restructed mathematics section.

config/en/mkdocs.yaml

@@ -40,7 +40,24 @@ extra_javascript:
 nav:
   - 'Welcome': index.md
   - 'Mathematics':
-    - 'Start': mathmematics/start.md
+    - 'Start': mathematics/start.md
+    - 'Calculus': 
+      - 'Limits': mathematics/calculus/limits.md
+      - 'Continuity': mathematics/calculus/continuity.md
+      - 'Differentation': mathematics/calculus/differentation.md
+      - 'Transcendental functions': 
+        - 'Inverse functions': mathematics/calculus/transcendental-functions/inverse-functions.md
+        - 'Exponential and logarithmic functions': mathematics/calculus/transcendental-functions/exponential-and-logarithmic-functions.md
+      - 'Extremes values': mathematics/calculus/extremes-values.md
+      - 'Concavity and inflections': mathematics/calculus/concavity-and-inflections.md
+      - 'Taylor polynomials': mathematics/calculus/taylor-polynomials.md
+      - 'Integration': mathematicse/calculus/integration.md
+      - 'Integration techniques': mathematics/calculus/integration-techniques.md
+      - 'Improper integrals': mathematics/calculus/improper-integrals.md
+    - 'Ordinary differential equations':
+      - 'First order differential equations': mathematics/ordinary-differential-equations/first-order-ode.md
+      - 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md
+      - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md
 
   - 'Physics':
     - 'Start': physics/start.md

docs/en/chemistry/start.md

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+# Mathematics
+
+Welcome to the chemistry page.

docs/en/mathematics/ordinary-differential-equations/first-order-ode.md

@@ -1,6 +1,6 @@
 # First-order differential equations
 
-## First order linear differential equations
+## First-order linear differential equations
 
 A first-order **linear** differential equation is one of the type
 

docs/en/mathematics/ordinary-differential-equations/second-order-ode.md

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+# Second-order ordinary differential equations
+
+For simplicity, all definitions and statements are for complex values functions and vector spaces over $\mathbb{C}$.
+
+## Linear second-order ODEs with constant coefficients
+
+Let $L[y] = f$ be given by
+
+$$
+L[y] = \ddot y + p \dot y + qy = f \qquad (*),
+$$
+
+with $f,p,q \in \mathbb{R}$. 
+
+*Definition*: the set of all solutions to $(*)$ is called the general solution.
+
+*Property*: if $y_1,y_2$ are both solutions to the homogeneous case $L[y]=0$ then $\forall c_1,c_2 \in \mathbb{R}$, $y=c_1y_1 + c_2y_2$ is a solution. 
+
+$$
+L[y] = L[c_1y_1 + c_2y_2] = c_1L[y_1] + c_2L[y_2],
+$$
+
+Then the consequence is that the general solution is a linear space.
+
+$(*)$ is said to have **resonance** if $f$ can be split into linearly independent terms of which at least one lies in the solution space of $(*)$.
+
+<br>
+
+### Solving homogeneous linear second-order ODEs with constant coefficients
+
+Therefore solving
+
+$$
+L[y] = \ddot y + p \dot y + qy = 0.
+$$
+
+Ansatz: let $y(t) = e^{\lambda t}$ with $\lambda \in \mathbb{C}$. Then
+
+$$
+L[y(t)] = \lambda^2 e^{\lambda t} + p \lambda e^{\lambda t} + q e^{\lambda t} = e^{\lambda t} (\lambda^2 + p \lambda + q) = 0,
+$$
+
+obtaining the characteristic equation $\Chi(\lambda) = \lambda^2 + p \lambda + q = 0$. If two roots $\lambda_1,\lambda_2 \in \mathbb{C}$ are found the solution space is 
+
+$$
+y(t) = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t}, \quad c_1,c_2 \in \mathbb{C},
+$$
+
+if instead one root $\lambda_1 \in \mathbb{C}$ is foundt the solution space is
+
+$$
+y(t) = (c_1 + c_2t) e^{\lambda_1 t}.
+$$
+
+*Proof*: will at some point be added.
+
+#### Example
+
+Let the homogeneous linear second-order ode be given by $\ddot y + 4 \dot y + 8y = 0$. Then the characteristic equation is given by $\Chi(\lambda) = \lambda^2 + 4\lambda + 8 = 0$ with solutions $\lambda_1 = -2 + 2i$ and $\lambda_2 = -2 - 2i$. Then the general solution is given by
+
+$$
+y(t) = c_1 e^{(-2 + 2i)1 t} + c_2 e^{(-2 - 2i) t}, \quad c_1,c_2 \in \mathbb{C},
+$$
+
+and we can write the real solution as
+
+$$
+y(t) = e^{-2t}\big(d_1\cos 2t + d_2 \sin 2t \big), \quad d_1,d_2 \in \mathbb{R}.
+$$
+
+<br>
+
+### Solving inhomogeneous linear second-order ODEs with constant coefficients
+
+*Theorem*: let $y_p$ be a particular solution to $(*)$. Then the general solution to $(*)$ is given by
+
+$$
+y = y_H + y_p,
+$$
+
+with $y_H$ the solution to the homegeneous case.
+
+*Proof*: let $y$ be a solution to $(*)$, then $L[y - y_p] = L[y] - L[y_p] = f - f = 0$. Therefore $y = (y - y_p) + y_p = y_H + y_p$.
+
+#### Method of variation of parameters
+
+We need the general solution to the homogeneous case 
+
+$$
+y_H(t) = c_1 y_1(t) + c_2 y_2(t), \qquad c_1,c_2 \in \mathbb{C}. 
+$$
+
+Ansatz: let $y_p(t) = c_1(t) y_2(t) + c_2(t) y_2(t)$, then taking the derivative of $y_p(t)$
+
+$$
+\dot y_p(t) = \dot c_1(t) y_2(t) + \dot c_2(t) y_2(t) + c_1(t) \dot y_2(t) + c_2(t) \dot y_2(t),
+$$
+
+we demand that $\dot c_1(t) y_2(t) + \dot c_2(t) y_2(t) = 0$. Then taking the second derivative of $y_p(t)$
+
+$$
+\ddot y_p(t) = \dot c_1(t) \dot y_2(t) + \dot c_2(t) \dot y_2(t) + c_1(t) \ddot y_2(t) + c_2(t) \ddot y_2(t),
+$$
+
+then we have for $(*)$
+
+$$
+\ddot y_p(t) + p \dot y_p(t) + q = c_1\big(\ddot y_1 + p \dot y_1 + q y_1\big) + c_2\big(\ddot y_2 + p \dot y_2 + q y_2\big) + \dot c_1 \dot y_1 + \dot c_2 \dot y_2 = f
+$$
+
+we demand that $\dot c_1 \dot y_1 + \dot c_2 \dot y_2 = f$. Then we can create a linear system of demands
+
+$$
+\begin{pmatrix} y_1 && y_2 \\ \dot y_1 && \dot y_2\end{pmatrix} \begin{pmatrix} \dot c_1 \\ \dot c_2 \end{pmatrix} = \begin{pmatrix} 0 \\ f \end{pmatrix},
+$$
+
+named the Wronskian and we can solve for $c_1(t)$ and $c_2(t)$ by integration.
+
+#### Ansatz method
+
+Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n and P_n$ polynomials of degree $\leq n$ and $\alpha \in \R$
+
+| Inhomogeneity | Particular solution |
+| ------ | --------------- |
+| $L[y] = P_n$ | $t^m A_n$ |
+| $L[y] = P_n e^{\alpha t}$ | $t^m A_n e^{\alpha t}$ |
+| $L[y] = P_n \cos \omega t$ | $t^m \big(A_n \cos \omega t + B_n \sin \omega t \big)$ |
+| $L[y] = P_n \sin \omega t$ | $t^m \big(A_n \cos \omega t + B_n \sin \omega t \big)$ |
+| $L[y] = P_n e^{\alpha t} \cos \omega t$ | $t^m e^{\alpha t} \big(A_n \cos \omega t + B_n \sin \omega t \big)$ |
+| $L[y] = P_n e^{\alpha t} \sin \omega t$ | $t^m e^{\alpha t} \big(A_n \cos \omega t + B_n \sin \omega t \big)$ |
+
+Choose $m \in \N \cup \{0\}$ as small as possible such that no term in the ansatz solves the homogeneous equation $L[y] = 0$.
+

docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md

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+# Systems of linear ordinary differential equations
+

docs/en/philosophy/start.md

@@ -0,0 +1,3 @@
+# Philosophy
+
+Welcome to the pilosophy page.

docs/en/physics/start.md

@@ -0,0 +1,3 @@
+# Physics
+
+Welcome to the physics page.