Suggest edit — Linear Constant-Coefficient Equation

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---
title: "Linear Constant-Coefficient Equation"
type: concept
related: [Linear Non Seperable Equation]
source: https://www.jemoka.com/posts/kbhlinear_constant_coefficient_equation/
confidence: high
status: active
---

Here it is:
\begin{equation} a\frac{dy}{dx} + by = c \end{equation}
For some constants \(a,b,c\). The name is pretty obvious, because we have constants and the highest power on everything is \(1\). Its first-order because the derivative is only the first-order derivative.
linear (diffeq) We technically call it &ldquo;linear&rdquo; because: if there are two possible solutions \(y_1(x)\) \(y_2(x)\), a linear combination \(Ay_1(x)+By_2(x)\) should also be a solution. Its &ldquo;linear&rdquo; because linear combinations work.
solving separable differential equations A separable differential equation means that we can separate the derivative by itself and separate its two components. For the example above, we have that:
\begin{equation} \frac{dy}{dx} = \frac{c-by}{a} \end{equation}
We can naturally separate this:
\begin{equation} \frac{a}{c-by}dy = dx \end{equation}
And then we can finally take the integral on both sides:
\begin{equation} \int \frac{a}{c-by}dy = \int dx \end{equation}
Wait wait wait but why is this possible? Why is it that we can separate a \(\frac{dy}{dx}\) such that \(dy\) and \(dx\) is isolatable? Remember:
\begin{equation} \frac{dy}{dx} = \lim_{h\to 0} \frac{y(x+h)-y(x)}{h} \end{equation}
no where is the differentials seperatable! Apparently Ted&rsquo;s undergrads didn&rsquo;t know this either. So here&rsquo;s a reading on it.
What if its non-seperable? See Linear Non-Seperable Equation