4.1: The exposition to conquer all exposition

Chapter 4 is entitled “Second-Order Equations.”

Section 4.1 is entitled “Definitions and Examples.” (With a section title like that, you know this summary is going to be filled with fun and happy times.)

This section is filled with lots of propositions and theorems and physics, so it won’t be so bad. Consider this the exposition to the next five sections you will see on this blog. Hooray! However, the thing about theorems is that there should be a proof alongside it, which means I have to write those up. Hooray.

A second order differential equation is similar to a first order differential equation, with the independent variable and the dependent variable and the first derivative, except (surprise, surprise) the second-order has a second derivative as well. Assuming we can solve for this second derivative, our equation would then have the form

The solution to this differential equation is what is called a twice continuously differentiable function y(t), where the form would be

An example of a second order equation would be Newton’s second law (F = ma), since acceleration is the second derivative of position. The force is usually a function of time, position, and velocity, so the differential equation would have the form

A special type of second order differential equations is linear equations, which have the form

As with the first order linear equations, the p and q and g are called coefficients. And, just like last time, y^’’ and y^’ and y must appear to first order. The right hand side of the equation (i.e. g(t)) is called the forcing term because it usually arises from an external force. If this forcing term is equal to zero, then the equation is considered homogenous. This means the homogeneous equation associated with our linear equation is of the form

Another example of a second order differential equation is the motion of a vibrating spring.

I drew a picture similar to the one in the book. The first position of the spring (marked with the (1) thingy on the right) is called spring equilibrium. This is where the spring will rest without any mass attached. The (2) position is called spring-mass equilibrium. This is the position x₀ where the spring is again in equilibrium with a mass attached. There are two forces acting on the spring: gravity and what is called the restoring force for the spring. We will denote the restoring force with R(x) since it depends on how far the spring has stretched. Equilibrium means the total force on the weight is zero, which means the equivalent force equation for this system is

In the (3) position, the spring is stretched even further and is no longer in equilibrium, which means the weight is probably moving. Allow me to remind you that velocity is the first derivative of position. (This will come in handy later.)

Now, for (2) we made a force equation for the system. Let’s do this for (3) as well. We still have gravity and the restoring force acting on the system, but now we also have what is called a damping force, which we will denote as D. The book defines this force as “the resistance to the motion of the weight due to the medium (air?) through which the weight is moving and perhaps to something internal in the spring” (138). The major dependence that this damping force has is velocity, so we can write this force as D(v). Finally, we’ll add in a function F(t) for any external forces acting on the system.

If we write acceleration as x^’’ (meaning the second derivative of the position), then we can write our total force on the weight, ma, as mx^’’. This means we can write our second order differential equation for the forces acting on this system as

Now, you might be asking “Well, how do we find restoring force?”

To which I say, “Physics!”

Hooke’s Law says that the restoring force is proportional to the displacement. This means the restoring force is

There’s a minus sign as to indicate the force is decreasing the displacement, which means k, which is known as the spring constant, is positive. Something to note about Hooke’s Law is that it is only valid for small displacements. So if we assume our restoring force follows Hooke’s Law, then our equation becomes

Just assuming that there is no external forces acting on the system, and that the weight is in spring mass equilibrium (position (2)). This means x is a constant, so its first and second derivatives are zero, which would make the damping force equal to zero. This means our equation would become

The book discusses units very quickly so I will too. The book uses the International System (as it should), where the unit of length is the meter (denoted m), the unit of time is the second (denoted s), and the unit of mass the kilogram (denoted kg). Everything other unit is derived from these fundamental units. For example, the unit of force is kg*m/s², which is known as the newton (denoted N).

Now you might be asking, “How do we find the damping force?”

The damping force always acts against velocity. This means the force takes on the form

μ is a nonnegative function of velocity. For some objects and for small velocities, the damping force is proportional to the velocity. This means that μ is a nonnegative constant, which is called the damping constant.

Some other examples for you (taken out of context of examples, but useful in many aspects):

Let’s now talk about the existence and uniqueness of second-order differential equations. They’re very similar to first-order. Also, just for your information, all of the theorems and propositions will be direct quotes from the book, since they explain them the best. Also, I’m going to use the theorem and proposition numbers they have in the book.

Theorem 1.17: “Suppose the functions p(t), q(t), and g(t) are continuous on the interval (α, β). Let t₀ be any point in (α, β). Then for any real numbers y₀ and y₁ there is one and only one function y(t) defined on (α, β), which is a solution to

and satisfies the initial conditions y(t₀) = y₀ and y^’(t₀) = y₁” (140).

The major difference between this theorem for second order differential equations and the related theorem for first order differential equations back in section 2.7 is that there is an initial condition needed for both the function y and the function y^’. Also note that this theorem can be sure a solution exists, and that it exists over the interval where the coefficients are defined and continuous.

Proposition 1.18: “Suppose that y₁ and y₂ are both solutions to the homogeneous, linear equation

Then the function

is also a solution to [this equation] for any constants C₁ and C₂” (141).

Here’s the proof for that:

Another definition for you: a linear combination of two functions u and v is a function of the form w= Au + Bv. A and B are just constants for this.

With this definition, our proposition can be stated by saying a linear combination of two solutions to a differential equation is also a solution to that differential equation.

Two functions u and v are linearly independent on an interval (α, β) if neither of them is a multiple of the other on that interval. If one is a constant multiple of the other on that interval, then they are said to be linearly dependent. For example, u(t) = t and v(t) = t² are linearly independent on the entire real plane (negative infinity to positive infinity). Linearly dependent functions would be u(t) = t and v(t) = 17t.

Theorem 1.23: “Suppose that y₁ and y₂ are linearly independent solutions to the equation

Then the general solution to [this equation] is

where C₁ and C₂ are arbitrary constants” (142).

Two linearly independent solutions, like the solutions in Theorem 1.23, form what is called a fundamental set of solutions.

So if we want to prove this theorem, we need to think about linear independence. Usually by simple observation, we can tell whether or not two functions are linearly independent. Sometimes that isn’t the case though. So we use something called the Wronskian. The Wronskian of two functions (let’s call them u and v) would be

You might be asking, “What does this tell us about anything?”

Here’s a proposition to answer this question for you.

Proposition 1.26 and 1.27 are about the results of the Wronskian and what that means for the homogenous differential equation. They are summed up in proposition 1.29, which I will quote for you now:

Proposition 1.29: “Suppose the functions u and v are solutions to the linear, homogeneous equation

in the interval (α, β). If W(t₀) ≠ 0 for some t₀ in the interval (α, β), then u and v are linearly independent in (α, β). On the other hand, if u and v are linearly independent in (α, β), then W(t) never vanishes in (α, β)” (144).

The proof of this proposition is the other two propositions that kind of work like exposition to this major proposition.

Tying everything back together, here’s the proof for Theorem 1.23:

One last thing to leave you with: When formulating initial value problems for second-order differential equations, we need to specify both a y(t₀) and a y^’(t₀). We had a theorem for that, but it applies to all second-order differential equations.

That’s all section 4.1 has to throw at you. I'll see you when I see you.