DiffEQ Updates

docs/math/diffeq/4-laplace.md

@@ -12,8 +12,6 @@ $$
 
 ## Section 4.2 - Properties
 
-This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/LaplaceTransforms.aspx).
-
 The Laplace Transformation is a linear transformation over functions in $\mathbb{R}[t]$. That is, given $a, b \in \mathbb{R}, f(t), g(t) \in \mathbb{R}[t]$, we know that
 
 $$
@@ -22,8 +20,6 @@ $$
 
 ## Section 4.3 - Inverse Laplace Transformation
 
-This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/InverseTransforms.aspx).
-
 Given $F(s)$, we define the Inverse Laplace Transformation as the following;
 
 $$
@@ -58,8 +54,6 @@ $$
 
 ## Section 4.5 - Laplace Transformation applied to IVPs
 
-This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx).
-
 **Theorem**. Given a function $f(t)$ with $C^n$ continuity, then
 
 $$
@@ -79,4 +73,34 @@ We can take the Laplace transformation of an IVP, solve for $Y(s)$, then take th
 
 ## Section 4.6 - Non-constant Coefficient IVPs
 
-This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx).
+If $f(t)$ is piecewise continuous on $[0, \infty)$, then $\lim_{s \rightarrow \infty} F(s) = 0$.
+
+**Definition**. A function $f(t)$ is said to be of exponential order $\alpha$ if there exists positive constants $T, M$ such that for all $t \geq T$, $|f(t)| \leq Me^{\alpha t}$.
+
+To check this, simply compute $\lim_{t \rightarrow \infty} \frac{|f(t)|}{e^{\alpha t}}$. If this is finite for some $\alpha$, then the function is of exponential order $\alpha$.
+
+## Section 4.7 - IVPs with Step Functions
+
+Recall that $\mathcal{L} \{u_c(t)f(t-c)\} = e^{-cs}F(s)$. Then, we can solve IVPs involving step functions.
+
+## Section 4.8 - Dirac Delta Function
+
+The Dirac Delta function has several properties. First, $\delta(t - a) = 0$ when $t \neq a$. Notably, though,
+
+$$\int_{\mathbb{R}} f(t) \delta(t - a) dt = f(a)$$
+
+Note that this is not an actual function, buy instead a *generalized function* or *distribution*, as several functions can express this property using infinite limits.
+
+Then, we can see that $\mathcal{L} \{\delta(t-a)\} = \int_0^\infty e^{-st} \delta(t-a) dt$ by definition. Then, applying the properties of the Delta function, $\mathcal{L} \{\delta(t-a)\} = e^{-as}$, given $a > 0$.
+
+## Section 4.9 - Convolution Integrals
+
+Consider two functions $F(s)$ and $G(s)$ such that $F(s) G(s) = H(s)$, of which we want to find an inverse Laplace transform.
+
+We define a *convolution integral* $(f*g)(t)$ as
+
+$$(f*g)(t) = \int_0^t f(t - \tau)(g - \tau) d\tau$$
+
+A unique property of this integral is that $(f*g) = (g*f)$.
+
+With this, we see that $\mathcal{L} \{f * g\} = F(s)G(s)$, or that $\mathcal{L}^{-1} \{F(s)G(s)\} = (f * g)(t)$.

docs/math/diffeq/5-systems.md

@@ -0,0 +1,7 @@
+# Section 5 - Systems of Differential Equations
+
+Sections 5.1-5.3 are review.
+
+## Section 5.4 - Systems of Differential Equations
+
+This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx).