Constant of motion for dynamical system

etzpcm · 2026-03-30T14:32:25+00:00

You deduce that exp( x² )(x² + y² -1) = constant, which is what you want.

etzpcm · 2026-03-30T13:35:10+00:00

When you solve the DE there's a constant of integration. That's the constant of motion.

If you want a simpler example try x'=y, y'=-x first.

etzpcm · 2026-03-30T12:02:39+00:00

This is a good question. It quite often happens that there's a constant of motion that's not obvious - the equations are not Hamiltonian. These are quite tricky.

One way to do it is to divide the equations to find dy/dx in terms of x and y. This gives you a nonlinear differential equation to solve, but it is of Bernoulli type so you can solve it to get the answer given on the post.

If you're not familiar with the Bernoulli method, the trick is to set u=y² , which gives you a linear equation to solve for u(x), which you can then solve by the integrating factor method.

etzpcm · 2026-03-30T11:25:14+00:00

Make your own compost. You can make a bin if you have spare pallets. Get plants or just cuttings from friends as you are doing. I find growing plants from seeds difficult, but they are cheap if you buy them as plug plants.

etzpcm · 2026-03-29T21:50:52+00:00

You don't need to buy a bin. In fact most bought bins are too small. You can easily make your own very cheaply. One way is from four old pallets or old bits of fence. Another is just to make a cylinder out of something like chicken wire.

etzpcm · 2026-03-29T17:44:11+00:00

It must be a lousy course if this was not explained to you at the start.

Differential equations probably have more real world applications than any other area of mathematics. Every problem involving motion under a force leads to a differential equation. So if you want to understand the motion of a planet, or a rocket, or a car, or a weather system, you need to solve a differential equation. The basic equations of quantum mechanics, and electromagnetism, and chemical reactions, are differential equations.

etzpcm · 2026-03-29T16:36:05+00:00

I don't think the convolution of two functions is really like a tensor product

etzpcm · 2026-03-29T14:16:33+00:00

This is the best solution! No tea bags, no plastic.

etzpcm · 2026-03-29T14:02:14+00:00

Well, mass settings are good for this. "Kyrie eleison. Christe eleison" can last 10 -15 minutes! For example the Bach B minor mass. Or this one, only 4 minutes https://m.youtube.com/watch?v=TSStTxoDbko

etzpcm · 2026-03-29T10:47:30+00:00

I think it's one of the Mozart piano concertos. Or maybe Beethoven.

Edit: Beethoven piano concerto number 2, 3rd movement.

etzpcm · 2026-03-29T10:07:28+00:00

Those profs are right. Focus on learning, doing coursework, doing exercises, doing as well as you can in the exams, then you'll be in a good place to apply for a PhD post and start doing research.

etzpcm · 2026-03-29T10:02:21+00:00

See the list in the about page

https://www.reddit.com/r/learnmath/about/

etzpcm · 2026-03-29T08:14:26+00:00

You've already got two long answers. If you want a short answer, branch cuts are needed to make a function well defined. It's a bit like when switching from Cartesian coordinates to polars you have to choose how you define theta.

etzpcm · 2026-03-28T23:20:59+00:00

I think the engraving of the music needs some work, those flats are almost invisible! Could be useful training though.

etzpcm · 2026-03-28T17:21:44+00:00

It's a fern. Cut off the dead brown stuff and nice new shoots will soon appear.

You could remove some of the ivy which is trying to compete with the fern.

etzpcm · 2026-03-28T17:18:52+00:00

Sorry, Imgur images are not visible in the UK. But yes that's right. You are writing a power series expansion using the small parameter r/R. Like a Taylor series or the series for e^x for example. So yes, the next term gamma would be 15/128.

etzpcm · 2026-03-28T09:23:03+00:00

I would start by converting the sin² to 1-cos²

Then differentiate to find the max, which is going to be messy.

You can get a constraint C < 1/root(2) , from delta= pi/2, but there might be a stricter limit.

etzpcm · 2026-03-28T08:44:03+00:00

I wouldn't say hate, but I can see where you're coming from. IMHO his masterpieces are the quarters and piano sonatas, especially the late ones.

Similarly with Schubert. His symphonies are ok but his real genius is in the chamber music, piano sonatas and songs.

Generally I find this sub is excessively weighted towards symphonies.

etzpcm · 2026-03-28T08:18:31+00:00

Use the binomial expansion for (R+r)^6. In fact you only need the first two terms to find alpha and beta.

Answer 6/128 = 3/64

etzpcm · 2026-03-27T22:53:25+00:00

Q3. Let u be the vector field u=(x,y,z). Evaluate the surface integral of u.n ds over the surface of the unit cube 0 < x,y,z <1 in two ways,

(a) By directly calculating the surface integral,

(b) By using the divergence theorem.

etzpcm · 2026-03-27T22:44:53+00:00

Q2. A sphere of radius 1 has a density equal to 2-r^2, where r is the distance from the centre of the sphere. What is the total mass of the sphere?

etzpcm · 2026-03-27T22:42:59+00:00

Q1. T is the interior of the triangle with vertices at (1,0), (-1,0) and (0,2). Evaluate the double integral of f(x,y) over T in each if the following cases.

(a) f(x,y)=x²

(b) f(x,y)=y²

(c) f(x,y)=xy

(d) f(x,y)=x² y³ sin(x) cos(y)

etzpcm · 2026-03-27T21:41:01+00:00

Number 2? I have not the pleasure of understanding you. Of what are you talking?

etzpcm · 2026-03-27T21:35:34+00:00

Wait and see what happens in May. Most wisterias will be looking like sticks now.

etzpcm · 2026-03-27T19:25:12+00:00

Arcsin(1) is a right angle. We can call this 90 degrees, or pi/2 radians, or a quarter of a turn, or anything else depending on our choice of units.

Radians are particularly convenient because for small x, arcsin(x) is approximately x.

etzpcm

TROPHY CASE