Completed the Laplace transform section.

docs/en/mathematics/ordinary-differential-equations/laplace-transform.md

@@ -1 +1,148 @@
 # The Laplace transform
+
+*Definition*: let $f: (0,\infty) \to \mathbb{R}$ be a piecewise continuous function that complies to the demand: $\exists s_0 \geq 0, \mu > 0: |f(t)| \leq \mu e^{s_0 t}$, then the **Laplace transform** $\mathcal{L}[f]$ is defined by
+
+$$
+\mathcal{L}[f](s) := \int_0^\infty e^{-st} f(t)dt = F(s),
+$$
+
+where $F(s)$ exists for all $s > s_0$.
+
+## Basic properties
+
+**Linearity**: if $f,g: (0,\infty) \to \mathbb{R}$ both have Laplace transforms, then $f + g$ also has a Laplace transform, and
+
+$$
+\mathcal{L}[f + g] = \mathcal{f} + \mathcal{g},
+$$
+
+on the interval where both are defined. 
+
+*Proof*: will be added sometime.
+
+If $c \in \mathbb{R}$ then $cf$ also has a Laplace transform and, 
+
+$$
+\mathcal{L}[cf] = c \mathcal{L}[f].
+$$
+
+**Shifting**: if $f$ has a Laplace transform $F$ on $(s_0,\infty)$ and $a \in \mathbb{R}$ then the function $g$ given by
+
+$$
+g(t) = e^{at} f(t)
+$$
+
+has a Laplace transform $G$ on $(\mathrm{max}(s_0 + a),0),\infty$, and
+
+$$
+G(s) = F(s-a)
+$$
+
+on this interval
+
+*Proof*: will be added sometime.
+
+**More shifting**: let $a>0$, if $f$ has a Laplace transform $F$ on $s_0, \infty$ then the function $g$ given by
+
+$$
+g(t) = \begin{cases} f(t-a) \qquad &\text{if } t \geq a, \\ 0 \qquad &\text{if } t < a \end{cases}
+$$
+
+has a Laplace transform G on $(s_0,\infty)$, and
+
+$$
+G(s) = e^{-as}F(s)
+$$
+
+on this interval.
+
+*Proof*: will be added sometime.
+
+**Scaling**: let $a > 0$. If $f$ has a Laplace transform $F$ on $(s_0, \infty)$ then the function $g$ given by 
+
+$$
+g(t) = f(at)
+$$
+
+has a Laplace transform G on $(as_0, \infty)$, and 
+
+$$
+G(s) = \frac{1}{a} F\Big(\frac{s}{a}\Big)
+$$
+
+on this interval.
+
+*Proof*: will be added sometime.
+
+**Derivatives**: if $f$ has a derivative $g$ having a Laplace transform $G$ on the interval $(s_0,\infty)$ then $f$ has a Laplace transform on the same interval, and
+
+$$
+G(s) = sF(s) - f(0).
+$$
+
+More generally, for higher derivatives we have (under analogous assumptions)
+
+$$
+\mathcal{L}[f^{(n)}](s) = s^n F(s) - \sum_{k=0}^{n-1} s^k f^{(n-1-k)}(0)
+$$
+
+*Proof*: for large enough $s$, the case $n=1$ follows by integration by parts
+
+$$
+\begin{align*} \mathcal{L}[f'](s) &= \int_0^\infty e^{-st} f'(t)dt, \\ &= \Big[e^{-st} f(t) \Big]_0^\infty + s\int_0^\infty e^{-st}f(t), \\ &= sF(s) - f(0) \end{align*},
+$$
+
+suppose $\mathcal{L}[f^{k}](s) = s^k F(s) - \sum_{r=0}^{k-1} s^r f^{(k-1-r)}(0)$ is true for $k \in \mathbb{N}$, then by assumption
+
+$$
+\begin{align*} \mathcal{L}[f^{k+1}](s) &= \int_0^\infty e^{-st} f^{(k+1)}(t)dt , \\ &= \Big[e^{-st} f^{(k+1)}(t) \Big]_0^\infty + s\int_0^\infty e^{-st}f^{(k)}(t), \\ &= s \mathcal{L}[f^{(k)}] - f^{(k)}(0), \\ &= s \Big(s^k F(s) - \sum_{r=0}^{k-1} s^r f^{(k-1-r)}(0)\Big) - f^{(k)}(0), \\ &= s^{k+1} F(s) - \sum_{r=0}^{k} s^r f^{(k-r)}(0) \end{align*}.
+$$
+
+## Examples
+
+**Solving a second order linear ODE**: with $y: \mathbb{K} \to \mathbb{R}$ given by 
+
+$$
+\ddot y + 4 \dot y + 4y = t \qquad \text{with } y(0) = 1 \text{ and } \dot y(0) = 0
+$$
+
+using the Laplace transform
+
+$$
+\begin{align*}
+\mathcal{L}[\ddot y + 4 \dot y + 4y](s) &= \frac{1}{s^2} \qquad \text{let } \mathcal{L}[y](s) = Y(s), \\ 
+s^2 Y(s) -s + 4(sY(s) - 1) + 4 Y(s) &= \frac{1}{s^2}, \\ 
+(s^2 + 4s + 4)Y(s) &= \frac{1}{s^2} + s + 4, \\ Y(s) &= \frac{s^3 + 4s^2 +1}{s^2(s+2)^2}, 
+\end{align*}
+$$
+
+then it may be solved with partial fraction decomposition and the inverse transform.
+
+**Solving a linear system of ODEs**: with $\mathbf{y}: \mathbb{K} \to \mathbb{R}^2$ given by
+
+$$
+\mathbf{\dot y}(t) = \begin{pmatrix} 5 & 1 \\ 1 & 5 \end{pmatrix} \mathbf{y}(t) \qquad \text{with } \mathbf{y}(0) = \begin{pmatrix} -3 \\ 7 \end{pmatrix} 
+$$
+
+using the Laplace transform
+
+$$
+\begin{align*} 
+    \mathcal{L}[\mathbf{\dot y}](s) &= \begin{pmatrix} 5 & 1 \\ 1 & 5 \end{pmatrix} \mathcal{L}[\mathbf{y}](s) \qquad \text{let } \mathcal{L}[\mathbf{y}](s) = \mathbf{Y}(s), \\ 
+    s \mathbf{Y}(s) - \mathbf{y}(0) &= \begin{pmatrix} 5 & 1 \\ 1 & 5 \end{pmatrix} \mathbf{Y}(s), \\ 
+    s \mathbf{Y}(s) + \begin{pmatrix} 3 \\ -7 \end{pmatrix} &= \begin{pmatrix} 5 & 1 \\ 1 & 5 \end{pmatrix} \mathbf{Y}(s), \\
+    \begin{pmatrix} 5 - s & 1 \\ 1 & 5 - s \end{pmatrix} \mathbf{Y}(s) &= \begin{pmatrix} 3 \\ -7 \end{pmatrix}, 
+\end{align*}
+$$
+
+using Cramer's rule
+
+$$
+\begin{align*}
+    &Y_1(s) = \frac{\mathrm{det}\begin{pmatrix} 3 & 1 \\ -7 & 5 - s \end{pmatrix}}{(5-s^2)-1}, \\
+    \\
+    &Y_2(s) = \frac{\mathrm{det}\begin{pmatrix} 5 - s & 3 \\ 1 & -7\end{pmatrix}}{(5-s^2)-1}
+\end{align*}
+$$
+
+both can be solved with partial fraction decomposition and the inverse transform.

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

@@ -5,13 +5,13 @@
 Let $\mathbb{K} = \mathbb{R} \lor \mathbb{C}$, $n \in \mathbb{N}$ and $A \in \mathbb{R}^{n \times n}$. Seek differentiable functions $y:\mathbb{R} \to \mathbb{K}^n$ such that 
 
 $$
-\mathbf{\dot y(t)} = A \mathbf{y}(t), \qquad t \in \mathbb{R}
+\mathbf{\dot y}(t) = A \mathbf{y}(t), \qquad t \in \mathbb{R}
 $$
 
 The solutions from a linear space, therefore the general solutions can be written as,
 
 $$
-\mathbf{\dot y(t)} = \sum_{k=1}^n c_k \mathbf{y}_k(t), \qquad c_k \in \mathbb{K}
+\mathbf{y}(t) = \sum_{k=1}^n c_k \mathbf{y}_k(t), \qquad c_k \in \mathbb{K}
 $$
 
 where $\{\mathbf{y_1}, \dots, \mathbf{y_n}\}$ is a linear independent set of solutions, i.e. the basis of the solutions space.
@@ -41,7 +41,7 @@ $$
 Let $I \subseteq \mathbb{R}$ be an interval, $\mathbf{f}: I \to \mathbb{R}$ continuous. Find functions $\mathbf{y}: I \to \mathbb{R}^n$ such that 
 
 $$
-\mathbf{\dot y}(t) = a \mathbf{y}(t) + \mathbf{f}(t), \qquad t \in I. \qquad (*)
+\mathbf{\dot y}(t) = A \mathbf{y}(t) + \mathbf{f}(t), \qquad t \in I. \qquad (*)
 $$
 
 *Theorem*: let $\mathbf{y}_p: I \to \mathbb{R}^n$ a particular solution for $(*)$ and $\mathbf{y}_H$ the general solution to the homegeneous system. Then the general solutions of the inhomogeneous system $(*)$ is given by