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Indigo5684
2025-09-30 13:19:25 -05:00
parent e1146e7b1c
commit a629d475d4
13 changed files with 26 additions and 24 deletions
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2
docs/math/diffeq/1-intro.md
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@@ -36,4 +36,4 @@ Note that $a_n(t)$ does not depeond on any derivative of $y$, so the presence of
This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx).
**Definition**. A directional field is the graph of a $t$ vs. $y(t)$, with vectors drawn at each point with a slope corresponding to $y'(t)$. Notably, each arrow will be pointed right (towards increasing $t$).
**Definition**. A directional field is the graph of a $t$ vs. $y(t)$, with vectors drawn at each point with a slope corresponding to $y'(t)$. Notably, each arrow will be pointed right (towards increasing $t$).

2
docs/math/diffeq/2-1st-order.md
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@@ -144,4 +144,4 @@ $$
\frac{1}{1-n}v' + p(x)v = q(x)
$$
After solving, be sure to rewrite in terms of $y$.
After solving, be sure to rewrite in terms of $y$.

12
docs/math/diffeq/3-2nd-order.md
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@@ -32,7 +32,7 @@ Thus, we allow the *charactaristic equation* of the differential equation to be
$$ ar^2 + br + c = 0 $$
# Section 3.2 - Real & Distinct Roots
## Section 3.2 - Real & Distinct Roots
This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx).
@@ -42,11 +42,11 @@ $$ y_1(t) = e^{r_1 t} $$
$$ y_2(t) = e^{r_2 t} $$
Thus,
Thus,
$$ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} $$
# Section 3.3 - Complex Roots
## Section 3.3 - Complex Roots
This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx).
@@ -120,7 +120,7 @@ Assume we have the differential equation as follows:
$$ y'' + p(t) y' + q(t) y = g(t) $$
The equivilent homogenous differential equation is
The equivilent homogenous differential equation is
$$ y'' + p(t) y' + q(t) y = 0 $$
@@ -156,7 +156,7 @@ Assume we have the differential equation as follows:
$$ y'' + p(t) y' + q(t) y = g(t) $$
The equivilent homogenous differential equation is
The equivilent homogenous differential equation is
$$ y'' + p(t) y' + q(t) y = 0 $$
@@ -164,4 +164,4 @@ For this method, we must have $y_1(t)$ and $y_2(t)$ known. Through a lot of math
$$
y_p = -y_1 \int \frac{y_2(t)g(t)}{W(y_1, y_2)} dt + y_2 \int \frac{y_1(t)g(t)}{W(y_1, y_2)} dt
$$
$$

3
docs/math/diffeq/4-laplace.md
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@@ -1,4 +1,5 @@
# Section 4 - Laplace Transformations
## Section 4.1 - Definition
This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx).
@@ -78,4 +79,4 @@ We can take the Laplace transformation of an IVP, solve for $Y(s)$, then take th
## Section 4.6 - Nonconstant Coefficient IVPs
This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx).
This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx).