Lecture notes · Differential Equations · Lecture 10

Continuation: Complex Characteristic Roots

MIT 18.03 Differential Equations, lecture 10, explained in plain English

Want this lecture to stick? In LessonBound you get checked practice questions for every part, can ask about anything you don't get, and can listen with your screen off. Free, no ads.

Learn this lecture free

Watch or read the original lecture: MIT OpenCourseWare.

Part 1 · 0:00–10:00

This part explains why complex roots of the characteristic equation produce oscillating solutions, and how to pull real solutions out of them.

Key ideas

Oscillations, such as swinging or vibrating, are linked to complex roots of the characteristic equation. The roots come in pairs of the form a plus or minus bi. 0:47
A complex solution can be split into its real part and imaginary part. Each part is a real solution: e to the at times cosine bt, and e to the at times sine bt. 2:21
We need real solutions because the equation has real coefficients that describe real quantities. 3:03
The other root, a minus bi, gives no new solutions. Switching the sign only flips the sign of one solution, and that sign gets absorbed into the arbitrary constant. 4:57
To check whether a complex expression is real, you can look for an imaginary part equal to zero, or check whether it stays the same when i is changed to minus i. 9:49

Words to know

characteristic equation: A special equation built from the numbers in the original problem. Its roots tell us what the solutions look like.
complex conjugates: Two numbers that are the same except that the sign in front of the imaginary part is flipped, like a plus bi and a minus bi.
imaginary part: The part of a complex number that is multiplied by i. If it is zero, the number is just a regular real number.
independent: Not a fixed multiple of each other. Two solutions that are independent give you genuinely different information.

Example

Think of a bell that rings and fades. The rings are the up-and-down part of the motion, and the fading is the shrinking part. The lecture says the complex root gives both at once, like a combined rise-and-fall pattern. The swing analogy here is my illustration, not something the lecture stated.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part shows how to tell when a complex-number solution describes a real answer, and how to rewrite that answer using sines and cosines.

Key ideas

A complex expression is real if it stays the same when you swap i for minus i. Swapping i for minus i turns a complex number into its complex conjugate. 10:17
When you apply this test to the solution, the two conditions on the constants turn out to be one condition. The second condition adds nothing new, so it is redundant. 12:40
So every real solution can be written with one arbitrary complex constant, C plus i times d. The second term's coefficient must be C minus i times d, its complex conjugate. 14:00
Many engineers and scientists like to write solutions in this complex form. To change it back to sines and cosines, you pull out the shared e to the a t factor and use the magic formulas. 15:15
The magic formulas are Euler's formula read backwards. Euler's formula turns a complex exponential into sines and cosines, and the backwards version turns a sine or cosine into complex exponentials. 19:35

Words to know

complex conjugate: The same complex number with the sign of its imaginary part flipped. You write it with a bar over the letter, like C bar.
coefficient: The number that multiplies another quantity. In this lesson, C and d are the coefficients that multiply the exponential terms.
Euler's formula: A rule that connects complex exponentials to sines and cosines. The lesson says you should know it well, and the magic formulas come from it.
redundant: Not needed, because it says nothing new. Here, the second condition is redundant because the first one already implies it.

Example

Think of a number and its mirror image: flip the imaginary part and you get the conjugate. If a picture looks exactly the same in the mirror, it is symmetric, and the lesson's test works the same way: if swapping i for minus i changes nothing, the answer is real. This is an everyday comparison I added, not something the lesson says.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how a spring-and-damper system behaves, and how its equation is written so that the oscillations and their frequency are easy to read.

Key ideas

Cosine and sine can be written using complex exponentials. Combining the two forms gives a real solution that still has two arbitrary constants, just like the familiar real form. 20:55
The lecturer rewrites the spring-and-damper equation with new names, 2p and omega nought squared, so the formulas are simpler to remember. The damping constant c and spring constant k become these new terms. 24:30
Oscillations happen only when the roots of the characteristic equation are complex. If the roots are both real, the system is over-damped and does not oscillate. 26:22
When p equals zero, there is no damping, so the motion is pure simple harmonic motion. In that case omega nought is the circular frequency of the oscillation. 28:37

Words to know

characteristic equation: A simpler equation built from the original one, where the letter r stands in for the derivative terms. Its solutions tell you how the system behaves over time.
circular frequency: How fast an oscillation repeats, measured in radians per second. A bigger number means faster back-and-forth motion.
dashpot: A device, like a shock absorber, that slows motion by pushing back in proportion to how fast something moves.
over-damped: A system with so much resistance that it slides back to rest without swinging back and forth.

Example

Think of a playground swing with no friction at the hinge: once pushed, it keeps swinging back and forth. Add a shock absorber that slows it, and the swing either stops after a few swings or, if it is very stiff, just creeps back to rest without swinging at all.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part explains when a damped spring keeps oscillating, what its motion looks like, and how damping changes how fast it wiggles.

Key ideas

Without damping, the motion is a simple back-and-forth wave described by cosine and sine. The lecturer calls the key number omega nought. 30:00
With damping, the motion still oscillates only if the damping is small enough. In the lecturer's terms, p must be less than omega nought, so the roots are truly complex. 31:18
When the motion oscillates with damping, the wiggles stay inside a shrinking exponential curve, e to the negative pt. That curve steadily pulls the swings smaller. 33:26
Because the swings keep getting smaller, the motion is not truly periodic. The lecturer calls its frequency a pseudo-frequency, meaning a fake or almost frequency. 36:28
The lecturer asks you to guess the effect of more damping on the pseudo-frequency. The answer given is that it goes down. 38:29

Words to know

damping: Friction or resistance that slowly takes energy out of a moving system, so its swings get smaller over time.
complex roots: The solutions to the equation that involve a square root of a negative number. In this lesson, they are what make the motion oscillate.
exponential: A curve that shrinks or grows at a rate that depends on its own size. Here, e to the negative pt shrinks steadily over time.
pseudo-frequency: The lecturer's name for the number of swings per second in the damped motion. The word pseudo means fake or almost, because the motion is not truly repeating.

Example

Think of a child on a swing that slowly loses energy to air resistance. Each swing is a little smaller than the one before, and the swing slows down over time. The back-and-forth still happens, but the height keeps shrinking.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–46:28

When a spring is damped, its motion becomes a wobble that slowly fades, and four numbers together describe exactly how it behaves.

Key ideas

When the roots become complex, the frequency changes to a new number called omega one, which is smaller than the original spring frequency. This new number sets how fast the motion bobs up and down. 40:00
The solution now looks like a fading curve, e to the minus pt, multiplied by a cosine wave. The fading part shrinks the motion over time, and the cosine part makes it swing back and forth. 40:35
Two successive times the motion crosses the middle line are separated by 2 pi divided by omega one. That gap is one full period of the bobbing motion. 42:02
Omega one depends only on the spring and the damping, so it is fixed by the equation itself. The amplitude and phase lag depend on the starting conditions instead. 43:46
The answer needs four numbers: p sets how fast the motion dies down, the phase lag shifts the wave in time, the amplitude sets the starting size, and omega one sets how fast it bobs. 45:47

Words to know

pseudo-frequency: A made-up-looking frequency. Here it is the bobbing speed of a damped spring, which is a bit slower than the spring's natural speed.
phase lag: How far a wave is shifted in time. It tells you where the wave is at the start compared with a plain cosine wave.
ODE: Short for ordinary differential equation. It is a rule that links a thing's motion to its speed and acceleration, here the spring and its damping.
amplitude: The size of the swing, or how big the wave is at its peak, before it starts to fade.

Example

Think of a playground swing that someone pushes once and then lets go. It swings back and forth at a steady rhythm while the swings get smaller and smaller. The rhythm is like omega one, and the shrinking swings are like the fading e to the minus pt part.

3 checked practice questions for this part are in the app.

← Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients · Lecture 13: Finding Particular Solutions to Inhomogeneous ODEs →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

LessonBound · Free lecture notes · Privacy · Terms · New York