Since working with second order equations builds on techniques as we go, we will first consider homogeneous equations. Our second order equations look like \(y'' + p(t)y' + q(t)y = g(t)\) and when they are homogeneous \(g(t)=0\) giving us \(y'' + p(t)y' + q(t)y = 0 \).

Homogeneous equations with constant coefficients look like \(\displaystyle{ ay'' + by' + cy = 0 }\) where a, b and c are constants. We also require that \( a \neq 0 \) since, if \( a = 0 \) we would no longer have a second order differential equation.

When introducing this topic, textbooks will often just pull out of the air that possible solutions are exponential functions. Unfortunately, at this point, you just need to take their word for it. Oh, you can check that what they say is correct. But where they came from will remain something of a mystery for now.

The idea is to find the roots of the polynomial equation \(ar^2+br+c=0\) where a, b and c are the constants from the above differential equation. This equations is called the characteristic equation of the differential equation. If we call the roots to this polynomial \(r_1\) and \(r_2\), then two solutions to the differential equation are
\( y_1 = c_1 e^{r_1t} \) and \( y_2 = c_2 e^{r_2t} \).
Usually we add these two solutions together and write it as one solution and write \( y = c_1 e^{r_1t} + c_2 e^{r_2t} \). This table summarizes these comments.

That's the basic idea. Now, if the roots of the polynomial are complex or repeated, there are slight variations of this idea. But in general, that's how this type of differential equation is solved.

Okay, here are some practice problems where the roots are real and distinct. Working these should get you familiar with the technique before moving on to more complex problems.

Introduction to Partial Differential Equations with Applications (Dover Books on Mathematics)

So, if the roots are real and repeated, we call this root \(r_1 = r_2 = r\). From the above discussion, we have one solution \( y_1 = c_1 e^{rt} \). The second solution is obtained by multiplying the first solution by t to get \( y_2 = c_2 te^{rt} \). (The reduction of order page contains an explanation of where this comes from.) So the combined solution is \( y = c_1 e^{rt} + c_2 te^{rt} \).

Okay, before we go on to discuss complex roots, let's watch this video discussing these two cases so far. This is a very in-depth discussion with great examples using a spring-mass-damped system. It will be well worth your time to watch it carefully.

Now let's work some practice problems with real, repeated roots.

When the roots in the above discussion are complex, they are in the form \(r_1 = \alpha + i\beta \) and \(r_2 = \alpha - i\beta \) and our solution looks like \(\displaystyle{ y = c_1 e^{(\alpha + i\beta) t} + c_2 e^{(\alpha - i\beta)t} }\). This form is fine but it will look better if we use Eulers Formula to put it into another form.

From infinite series we know that
\(\displaystyle{ e^x = \sum_{n=0}^{\infty}{ \left[ \frac{x^n}{n!} \right] }, ~~~ -\infty < x < \infty }\).
This is the Taylor series for \(e^x\) about \(x=0\).
If we let \( x = it \), then this sum becomes
\(\displaystyle{ e^{it} = \sum_{n=0}^{\infty}{\frac{(it)^n}{n!}} = \sum_{n=0}^{\infty}{\frac{(-1)^nt^{2n}}{(2n)!}} + i\sum_{n=1}^{\infty}{\frac{(-1)^{n-1}t^{2n-1}}{(2n-1)!}} }\)
Notice that we have separated the real and imaginary parts of the series (see note below). The first series is the Taylor series for \(\cos(t)\) about \(t=0\) and the second series is the Taylor series for \(\sin(t)\) about \(t=0\). So this gives us Euler's Formula
\( e^{it} = \cos(t) + i \sin(t).\)

Now that we have Euler's Formula, we can solve homogeneous equations with constant coefficients when the characteristic equation has complex roots, just as we did when the roots were real and not equal.

Note - When substituting \( x=it \) we have moved from the real domain to the complex plane. For the sake of argument we will assume this jump is valid without proof since this discussion is only meant to give you a feel for Euler's Formula.

Here is a video explaining this in more detail.

After some manipulation, we can write this solution as \( y = e^{\alpha t}( A\cos(\beta t) + B\sin(\beta t) ) \). This video shows the details. It is a continuation of the spring-mass-damped system video.

Okay, let's work some practice problems with complex roots.

The following table summaries the three cases based on the types of roots. In this table, \(y\) is a function of \(t\) and \(a\), \(b\) and \(c\) are constants. We use the variable \(r\) in the characteristic (sometimes called auxiliary) equation but you may see other instructors use different variables. Check with your instructor to see what they expect.
Also, some instructors use hyperbolic functions in their answers. From our experience, this is pretty rare but there are several practice problems with video solutions that use this notation.

Higher-order linear equations work exactly like first and second-order, just with additional roots. Here are some practice problems to demonstrate this. There is nothing new here, just more terms in the equations. You need to factor into linear and/or quadratic terms and apply the techniques described above.