Ey"+(3+x)y'+xy=0, 0<x<1, 0<E<<1, subject to y(0)=20, y(1)=64. Find a leading approximation to the inner and outer solutions and match them.
However, I can't seem to match the two solutions, because when E tends to 0 in the inner solution, the inner solution tends to 20. While, my outer solution tends to 27e as E tends to 0.
I stretched x=Ex, and since the boundary layer is near x=0, i used y(1)=64 for the outer solution, and y(0)=20 for the inner.
Please help if anyone knows about this.
Oh, actually I was wanting an easy way to get to reduced row echelon form, not just the normal echelon form. It's just sometimes driving me crazy. It's always like I've reduced an element in the matrix to zero, but because of another row operation, the zero becomes a number again, and I'll be going around in circles.
And this row reduction to reduced row echelon form, is for me to find eigenvectors.
I think this is the place where I found it.
Unit Tangent is easy to get. But when I get to unit normal, it kinda seems too difficult sometimes, particularly when you're trying to find T'.
Or when I try to use the tangential and normal accelerations (a=aT'+v^2KN) formula, that can get very hard too. (where a is the acceleration, T is the unit tangent vector, v is the velocity, K is the curvature and N is the unit normal).
So does anyone know a simpler way of doing this?
If x=y=100 and increasing at the rate of 1 per second, z=200, decreasing at the rate of 2 per second. Is R decreasing or increasing and at what rate?
I don't know how to do this question because R is not defined as R(x,y,z) but rather as 1/R. What do I do with this?
Btw I still don't know how to use the math tags. I tried encasing intag but the output was [math?] without my equations.
Apply Gram Schmidt process to the power functions 1,t,t^2,t^3,t^4,t^5 to obtain the orthogonal bases of P1,P2,P3 and P4.
I don't think I know how to apply Gram-Schmidt for this one. What I know is how to apply Gram Schmidt for bases eg:
However when it comes to power functions, I'm stuck.
This is another part of the vector space and subset question.
Thanks for helping.
Suppose P represents the vector space of all real polynomials and that for n=0,1,2... Pn denotes the subspace of P consisting of all polynomials of degree at most n. Suppose that the inner producr on P is defined by:
(f,g)=S-1 to 1 f(t)g(t) dt (where S is the integral sign, I don't know how to use the tags yet)
Explain why the power functions 1,t,t^2, t^3, ..., t^n form a basis of Pn.
Some of my friends have tried solving this question by using some really complex methods (Cauchy-Schwarz inequality etc) that are 2-3 pages long! I don't agree with them. I think, since 1,t,t^2, t^3,..., t^n is a polynomial, and P represents the vector space of all real polynomials, so, by logic the functions form a basis of Pn, right?
Btw, does anyone know any easy way to row reduce a complex matrix to find eigenvectors ie. :
F4=1/2 (1 1 1 1)
1 i -1 -i
1 -1 1 -1
1 -i -1 i
1/2 times the matrix 4 X 4. I hope you can read it.
Eigenvalues=1,-1, i, 1
After doing eg F4-I, I have to do row reduction to find eigenvectors, and the problem is, it's tedious.
Prove that if norm(u X v)=0 then u.v=a norm(v)^2 where a is any constant.
I know that both u and v are parallel vectors and that norm(u X v) sin x = 0
Then I got stuck.
Anyone knows any good resources for me to learn linear algebra (preferably from the basics, with exercises and answers)? I'm finding it v hard to understand this topic.
The answer seems like the one I had above. Anyway, thanks for showing me another method to solve DEs.
The next question goes: y''+y'+2y=t^2+e^-t+cos t.
Do I just find y''+y'+2y'=cos t? I've already calculated for t^2 and e^-t as was shown by numen. Then after I get all these, I just add them together with A and B. Is it right?
Maths HAS loads of work, particularly when you get to advanced maths such as what I'm doing now. Actually there is a much harder maths out there called Fourier Analysis and Signal Processing. Last year I heard that no one passed that unit!
As most people here have said, physics and maths go hand in hand. Even biology needs maths (to model the behaviour of enzymes for example). One of my biology lecturers always complains that students in his class can't master basic calculus!
Does anyone know where to find worksheets on Linear Algebra and Matrices?
(projection theory, vector spaces, kernel, inner product, cauchy-schwarz, gram-schmidt, fourier and haar bases for linear algebra. Hadamard matrices, real and imaginary matrices, adjoints, unitary matrices etc for matrices)
I have a feeling that these sections are the hardest of all in all of the maths that I've learnt this semester (I learnt about vectors, partial derivatives, multiple integrals, vector calculus and this section) because it is so abstract, and worse, I don't have any references to them! That's why I'm looking for resources for me to practise these problems if anyone knows any.