Professor Blythe's Problem

Without looking up, Professor Bryce said hoarsely, "The problem is on the blackboard. You have thirty minutes."

Li Mo said, "All right."

There was only one problem on the blackboard.

It was a second-order differential equation involving a linearized approximation of orbital perturbations.

Wayne looked at it for five seconds before turning away. He knew it had little to do with him, so he decided to keep pretending to be a gentleman.

Anna was thinking hard about it, but the problem was far too difficult. It went well beyond what Mr. Harrison had taught in class, and she couldn't think of a way to approach it.

For Li Mo, it was a very basic ordinary differential equation.

Very simple.

What he was wondering was whether he should pretend to think it over. That might seem more reasonable.

But after glancing at Professor Bryce, Li Mo decided to be more direct.

The professor was clearly in a terrible mood. Perhaps giving him a little shock would help him out of it.

Li Mo stepped forward, picked up the chalk, and wrote down his calculations.

Anna and Wayne exchanged a glance.

That was way too fast. Less than a minute had passed since the professor finished speaking, and he already knew how to solve it?

Li Mo quickly wrote down the characteristic equation, followed by the general solution. Finally, he substituted the initial conditions and gave the expression directly.

Two minutes later, Li Mo said, "Professor, I'm done."

Professor Bryce remained silent.

After a long while, he slowly raised his head and glanced at the blackboard. Then he said softly, "Good. You got it right. Hmm... Let me give you another problem."

Li Mo said, "All right."

Professor Bryce held out the sheet of paper in his hand. "The problem is on here. Take a look."

Li Mo took the paper from the professor.

It was crumpled and covered in corrections. He could barely make out the problem.

It was a problem involving tidal calculations.

More precisely, it asked for a prediction of the time and height of the highest tide at Mendel Port on the coming autumnal equinox.

To solve the problem, the first half of the page listed formulas for gravitational potential energy and tidal observation constants from the past few years at Mendel Port, along with quite a few calculations.

To arrive at the final result, he had to use a trigonometric series, commonly known as a Fourier series, to perform harmonic analysis on the periodic changes in the satellite's gravitational potential energy.

Li Mo frowned. The difficulty of this problem... was entirely different from the previous one.

Based on what he knew of the mathematical level of this world, this was definitely not a problem a student could solve.

To put it bluntly, if the professor gave students a problem like this, he never intended to take on an assistant at all, did he?

This was clearly a bottleneck in the professor's own research... Even he was stuck on this calculation...

Li Mo scratched his head.

Though he didn't know what the professor was up to, he could solve it, at least.

It was just a lot of calculation and would take some time.

He couldn't solve this one in his head.

Li Mo said, "Professor, I need to borrow some scratch paper and a pen. It'll probably take some time."

Professor Bryce nodded blankly and pointed to the desk nearby.

Li Mo went over to the desk and pushed aside some of the star charts scattered across it, clearing a space.

Before starting the calculations, he closed his eyes and adjusted his breathing.

He silently said "quiet" and entered the focused state.

Without focus, even he would probably need more than a day to work out the problem. The amount of calculation really was enormous.

But in the focused state, he was basically a human computer...

He would be much more efficient.

Li Mo established the fundamental frequency, then wrote down the integral expressions for the Fourier coefficients.

The first step was separation of variables, splitting the gravitational potential energy into two independent periodic functions...

While Li Mo worked, Anna went over to the blackboard and pondered.

Wayne stood around the room for a while. Feeling bored, he went over to Anna and asked, "Anna, can you understand what Li Mo wrote? Explain it to me."

Anna thought, I only half understand it myself. In this first line, Li Mo seems to have assumed directly that the solution took the form of an exponential function, then substituted it into the original equation. That turns the original differential equation into an algebraic equation. Once you solve for the roots of that algebraic equation, you can write out the fundamental set of solutions directly...

Wayne suddenly felt sleepy.

It felt as though the words went in through his ears and flowed right back out without passing through his brain. How amazing...

Time ticked by.

At the desk, Li Mo had filled three whole sheets of scratch paper.

He had used lots of methods to reduce the amount of calculation.

For example, product-to-sum formulas could turn complicated product terms into simple addition and subtraction terms, greatly simplifying integration.

When dealing with the long-period modulation effects caused by irrational frequency ratios, he also used a Taylor series expansion with approximate truncation.

Even with so many little tricks, it still took him an hour and a half to finish.

After finishing,

Li Mo checked it over again.

He confirmed there were no errors or problems.

He raised his head, ready to leave the focused state.

But just then, a familiar sight flashed across his field of vision...

To his left, in a corner among a pile of discarded gears, he saw a familiar white thread.

Li Mo couldn't help taking another look.

The thread seemed to originate from a small object resting on top of a stack of documents...

He tried to get a closer look, but his eyes suddenly felt dry.

With no other choice, he blinked.

When he opened his eyes again, he had already left the focused state.

To the naked eye, the object resting on the documents seemed to be nothing more than an ordinary little ornament.

It looked like a dark bronze metal sphere with some scratches on its surface and a stand underneath, rather like a miniature globe.

To the naked eye, there really was nothing about it that suggested it was a mystical artifact.

Li Mo thought, Could this be an unexpected bonus?

The professor left it lying around, so perhaps he doesn't know it's a mystical artifact himself.

If it really does contain mystical power, I should find a chance to see if I can get the professor to give it to me.

As for fooling the old man...

More importantly, Li Mo had discovered a new function:

It turned out that focus could not only improve his calculation efficiency, but also help him find overlooked treasures!

It looked like he should use focus more often whenever he had nothing else to do.

He loved finding overlooked treasures!

Li Mo put those thoughts aside.

He could leave that for now. There would be plenty of time to study it later.

For the moment, he had to deal with the professor first.

After putting his scratch paper in order, he brought it all over to Professor Bryce.

The professor was still slumped in the same dejected posture as before, except his head hung even lower, as though he hadn't shaken off a hangover.

Li Mo held out the papers and said, "Professor, I solved it."

Professor Bryce didn't move.

Li Mo said again, "Professor, I solved it."

Only then did Professor Bryce slowly raise his head, his vacant gaze gradually coming into focus...

It was the first time he had looked Li Mo in the eye since Li Mo entered.

As though he could barely understand what he was being told, the professor asked, "What... did you say?"

Li Mo really didn't know what else to do. He gave a wry smile and said, "Professor, I solved the problem. Take a look."

The professor tilted his head.

He had understood what Li Mo said.

But his instinctive reaction was:

How was that possible?

Hadn't this kid Li Mo bought the problem, just like Matthew? Did he actually understand mathematics? He could even solve this problem?

Professor Bryce thought:

If Li Mo could solve this problem in such a short time, then he ought to call Li Mo his teacher and become Li Mo's assistant!

So he didn't believe it. It was impossible.

Absolutely-impossible.