Graph the two basic solutions of the differential equation and several other solutions. What fea

Graph the two basic solutions of the differential equation...

Key Concepts

Oscillatory Behavior

Oscillatory behavior in the solution arises when sinusoidal functions, such as sine and cosine, are part of the solution due to the imaginary component of the roots. This behavior is characterized by periodic fluctuations, and the frequency of oscillation is determined by the magnitude of the imaginary part of the roots.

Exponential Decay Factor

An exponential decay factor, usually of the form e^(?x) where ? is negative, appears in the solution of differential equations with complex roots with negative real parts. It ensures that all solutions tend to decrease in amplitude over time, causing the oscillations to eventually die out.

Characteristic Equation

The characteristic equation is a quadratic equation derived by assuming an exponential solution of the form y = e^(rx) for the differential equation. Solving this equation yields the values of r, which determine the behavior of the solution. It is foundational in transforming differential equations into algebraic problems.

Second-Order Linear Differential Equations

These are differential equations involving the second derivative of an unknown function. They have the general form ay'' + by' + cy = 0, and their solutions often depend on the roots of the associated characteristic equation. Their study is essential in describing many physical systems, including mechanical oscillators and electrical circuits.

Complex Roots

When the characteristic equation yields complex roots, the general solution of the differential equation involves a combination of exponential decay and sinusoidal functions. This leads to solutions that exhibit oscillatory behavior while also decaying or growing, depending on the sign of the real part of the roots.

Transcript

00:02 They wanted us to plot the two fundamental solutions of this differential equation, y double prime plus 2y plus 2y plus 2y plus 2y equals 0.

00:12 I taught vibrations classes for a long time, so i'd like to think of the second order differential equations in terms of vibration parameters.

00:22 So we can find, and we'll talk about, they talk about that later in the chapter, but this is the natural frequency squared, so natural frequency is squared to 2.

00:30 And this is two times the damping ratio times the natural frequency.

00:35 So that gives us natural frequency is square to two, and the damping ratio is 1 over square to 2.

00:42 And so we define, so the one solution is a, e to the minus zeta, omega, and t, cosine, omega, n times a square root of 1 minus aeta square t.

00:54 And in this case, that just comes down to a, e to the minus t, cosine of t.

01:01 Likewise, the second solution looks exactly the same, different constant obviously, and a sign function here instead of a cosine.

01:08 And that one turns out to be b, e to the minus t, sine of t.

01:12 And so they asked us to plot these things and comment on them.

01:17 Well, first of all, we noticed that, let's see here, this thing, this is the first one, y1.

01:26 And i just set a and a b equal to 1 here.

01:31 Obviously these would scale as these things change...

Please give Ace some feedback