## First-Degree Homogeneous Linear Equations with Constant Coefficients

Now that we solved one differential equation, let’s try a wider class: “first-degree homogeneous linear equations with constant coefficients”. Let’s break this down.

- First-degree: only involves the undetermined function and its first derivative
- Homogeneous: harder to nail down, but for our purposes it means that every term involves the undetermined function or its first derivative
- Linear: no products of the undetermined function or its first derivative with each other
- Constant Coefficients: only multiplying the undetermined function and its derivative by constants

Putting it all together, this means that our equation must look like this:

We can divide through by to assume without loss of generality that (if then the equation isn’t very interesting at all).

Now let’s again assume that is analytic at , so we can write

and

So our equation reads

That is, for all , and is arbitrary. Just like last time we see that multiplying by at each step gives a factor of . But now we also multiply by at each step, so we find

And indeed, we can rewrite our equation as . The chain rule clearly shows us that satisfies this equation.

In fact, we can immediately see that the function will satisfy many other equations, like , , and so on.

[…] and Cosine Now I want to consider the differential equation . As I mentioned at the end of last time, we can write this as and find two solutions — and — by taking the two complex […]

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[…] can be used to express analytic functions. Then I showed how power series can be used to solve certain differential equations, which led us to defining the functions sine and cosine. Then I showed that the sine function must […]

Pingback by Pi: A Wrap-Up « The Unapologetic Mathematician | October 16, 2008 |

Congratulation Sir. The power series has been commonly used to solve second order ODE. But, here you have explored to solve the first order ODE. I obtain so many information from you.

Comment by rohedi | November 29, 2008 |