Houjun Liu

Linear Constant-Coefficient Equation

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 “linear” 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 “linear” 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’s undergrads didn’t know this either. So here’s a reading on it.

What if its non-seperable? See Linear Non-Seperable Equation