Add sixth semester Ecuaciones Diferenciales II notes

MAC/6th Semester/Ecuaciones Diferenciales II.md

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+---
+type: 🏫
+tags:
+  - MAC/6/ED2
+---
+
+# Integral Transforms
+
+An [[Integral Transform|integral transform]] is an operator that takes a set, a function to be operated and another _specific_ function (called the integral kernel), returning one more function that's obtained through integration. It's basically just integrating the product of a function by the integral kernel over a given set. 
+
+## Laplace Transform
+
+The [[Laplace Transform|Laplace transform]] is an integral transform whose integral kernel is $e^{-st}$ and integration interval is $[0, \infty)$. 
+
+We can obtain the [[Laplace Transform of Simple Functions|Laplace transform of varios simple functions]] to facilitate other more complex transforms. Similarly, there are several [[Fundamental Theorems of the Laplace Transform|fundamental theorems]] that we can also use to our convenience. We can use one of them to easily obtain the Laplace transform of a piecewise function after having it [[Expressing a Piecewise Function with Shifted Unit Step Functions|rewritten in terms of shifted unit step functions]].
+
+We can guarantee the existence of a Laplace transform by [[Laplace Transform Conditions of Existence|means of the consequences of two theorems]].
+

Mathematics/Expressing a Piecewise Function with Shifted Unit Step Functions.md

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+---
+date: 2024-02-13
+type: 🧠
+tags:
+  - MAC/6/ED2
+---
+
+**Topics:** [[Piecewise Function]]
+
+---
+
+_**(theorem)**_
+
+Let $f(t)$ be a [[Piecewise Function|piecewise function]] of the following form:
+
+$$
+f(t) =
+\begin{cases}
+g(t) & \text{if } 0 \leq t < a \\
+h(t) & \text{if } a \leq t < b \\
+k(t) & \text{if } b \leq t
+\end{cases}
+$$
+
+We can write such a function as follows, by using the [[Shifted Unit Step Function|shifted unit step function]] $U$:
+
+$$
+\begin{align*}
+f(t) &= g(t) - g(t) U(t-a) + h(t) U(t-a) - h(t) U(t-b) + k(t) U(t-b) \\[0.5em]
+&= g(t) + [h(t) - g(t)] U(t-a) + [k(t) - h(t)] U(t-b)
+\end{align*}
+$$

Mathematics/Fundamental Theorems of the Laplace Transform.md

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+---
+date: 2024-02-13
+type: 🧠
+tags:
+  - MAC/6/ED2
+---
+
+**Topics:** [[Laplace Transform]]
+
+---
+
+_**(theorems)**_
+
+There are several theorems that prove to be fundamental when working with [[Laplace Transform|Laplace transforms]]:
+
+$$
+\begin{align*}
+\mathcal{L}\{f(at)\} &= \frac{1}{a} F \left( \frac{s}{a} \right)  & \quad \text{(t. 9)} \\[1em]
+\mathcal{L}\{f^{n}(t)\} &= s^{n} F(s) - s^{n-1} f(0) - \dots - f^{n-1} (0) & \quad \text{(t. 10)} \\[1em]
+\mathcal{L}\{e^{at}f(t)\} &= F(s-a) & \quad \text{(t. 11)} \\[1em]
+\mathcal{L}\{f(t-a)U(t-a)\} &= e^{-as} F(s) & \quad \text{(t. 12)} \\[1em]
+\mathcal{L}\{t^{n}f(t)\} &= (-1)^{n} \frac{d^{n} F(s)}{ds^{n}}& \quad \text{(t. 13)} \\[1em]
+\mathcal{L}\{f(t)g(t)\} &= F(s) G(s) & \quad \text{(t. 14)} \\[1em]
+\end{align*}
+$$
+
+…where $U(t-a)$ is the [[Shifted Unit Step Function|shifted unit step function]]:
+
+$$
+U(t-a) =
+\begin{cases}
+0 & \text{if } 0 \leq t \leq a \\
+1 & \text{if } t > a
+\end{cases}
+$$
+
+> [!info]- Theorem Numbering
+> The theorems are numbered according to my [[Ecuaciones Diferenciales II]] course.

Mathematics/Integral Transform.md

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+---
+date: 2024-02-01
+type: 🧠
+tags:
+  - MAC/6D
+---
+
+**Topics:** [[Integral Transform]]
+
+---
+
+An **integral transform** is an [[Operator|operator]] that is derived from other [[Function|functions]] though a [[Definite Integral|definite integral]]. They allow us to switch from a reference system to another, with the advantage of being able to go back to the original system (this ability being the inverse transform).
+
+Instances of integral transforms include the [[Laplace Transform|Laplace transform]].
+
+_**(definition)**_
+
+Formally, an **integral transform** is an operator that associates a new function to a given [[Set|set]] through integration with respect to a given parameter:
+
+$$
+T\{f(t)\} = F(s) = \int_{a}^{b} K(t,s) f(t) \ dt 
+$$
+
+…where:
+
+- $K(t,s)$ is the _integral kernel_ of the transform
+- $(a,b)$ is the interval that is commonly infinite
+- $F(s)$ is the _transform_ of the function $f$
+
+Do note that integral transforms are [[Linear Transformation|linear transformations]]. 

Mathematics/Laplace Transform Conditions of Existence.md

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+---
+date: 2024-02-19
+type: 🧠
+tags:
+  - MAC/6/ED2
+---
+
+**Topics:** [[Laplace Transform]]
+
+---
+
+Concerning [[Laplace Transform|Laplace transforms]], we can guarantee their existence by means of the consequences of the following theorems. 
+
+_**(theorem, 8)**_
+
+Note that, as $t$ increases, the growth of the [[Complex Number Module|module]] of the function $f(t)$ is not greater than that of an [[Exponential Function|exponential function]]. That is, there exists $M > 0$ and $s_{0} \geq 0$ such that, for every value of $t$:
+
+$$
+\left| f(t) \right| < Me^{s_{0}t}
+$$
+
+We call $s_{0}$ the _growth exponent_ of the function $f(t)$. This condition guarantees the existence of the Laplace transform. 
+
+_**(theorem)**_
+
+If $f(t)$ is [[Piecewise Continuous|piecewise continuous]] in every finite set $0 < t < M$ of exponential order $s_{0}$ for $t > M$, then the Laplace transform $F(s)$ exists as long as $s_0 > 0$.

Mathematics/Laplace Transform of Simple Functions.md

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+---
+date: 2024-02-06
+type: 🧠
+tags:
+  - MAC/6/ED2
+---
+
+**Topics:** [[Laplace Transform]]
+
+---
+
+_**(theorems)**_
+
+The [[Laplace Transform|Laplace transform]] of simple functions like $f(t) = c, f(t) = t$, etc. is as follows:
+
+$$
+\begin{align*}
+\mathcal{L}\{c\} &= \frac{c}{s} & s>0 & \qquad \text{(theorem 1)} \\[1em]
+\mathcal{L}\{t^n\} &= \frac{n!}{s^{n+1}} & s>0 & \qquad \text{(theorem 2)} \\[1em]
+\mathcal{L}\{\sin(at)\} &= \frac{a}{s^{2} + a^{2}} & s>0 & \qquad \text{(theorem 3)} \\[1em]
+\mathcal{L}\{\cos(at)\} &= \frac{s}{s^{2} + a^{2}} & s>0 & \qquad \text{(theorem 4)} \\[1em]
+\mathcal{L}\{e^{at}\} &= \frac{1}{s-a} & s>0 & \qquad \text{(theorem 5)} \\[1em]
+\mathcal{L}\{\sinh(at)\} &= \frac{a}{s^{2}-a^{2}} & s>0 & \qquad \text{(theorem 6)} \\[1em]
+\mathcal{L}\{\cosh(at)\} &= \frac{s}{s^{2}-a^{2}} & s>0 & \qquad \text{(theorem 7)}
+\end{align*}
+$$
+
+> [!info]- Theorem Numbering
+> The theorems are numbered according to my [[Ecuaciones Diferenciales II]] course.

Mathematics/Shifted Unit Step Function.md

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+---
+date: 2024-02-19
+type: 🧠
+tags:
+  - MAC/6/ED2
+---
+
+**Topics:** [[Unit Step Function]]
+
+---
+
+_**(definition)**_
+
+The **shifted [[Unit Step Function|unit step function]]** is defined as:
+
+$$
+U(t-a) =
+\begin{cases}
+0 & \text{if } t \leq a \\
+1 & \text{if } t > a
+\end{cases}
+$$
+
+As the name suggests, it's just a unit step function that has been shifted from $0$ to $a$.

Mathematics/Unit Step Function.md

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+---
+date: 2024-02-19
+type: 🧠
+tags:
+  - MAC/6/ED2
+---
+
+**Topics:** [[Function]]
+
+---
+
+_**(definition)**_
+
+The **unit step function** is defined as:
+
+$$
+u(t) =
+\begin{cases}
+0 & \text{if } t \leq 0 \\
+1 & \text{if } t > 0
+\end{cases}
+$$
+
+In other words, the unit step function is a [[Function|function]] that returns $0$ when its argument is non-positive, and $1$ when it is positive. Sometimes $<, \geq$ are used instead of $\leq, >$.
+
+We can [[Shifted Unit Step Function|shift this function]] so that it's centred around a number other than $0$.