Intereting Posts

A familiar quasigroup – about independent axioms
Show that $ \Lambda:a\mapsto \Lambda_a $ defines a linear map from $ l^1(\mathbb{N}) $ to $ c^0(\mathbb{N})^* $
Is Fourier transform characterized by its diagonalization properties?
Question about the proof that the Hilbert Cube is compact.
About functions of bounded variation
Explain why calculating this series could cause paradox?
If every vector is an eigenvector, the operator must be a scalar multiple of the identity operator?
Interior of arbitrary products
Natural way of looking at projective transformations.
For every cardinal $\kappa$, $\kappa^+$ is regular
Find $a, b, c, d \in \mathbb{Z}$ such that $2^a=3^b5^c+7^d$
Counting problem: generating function using partitions of odd numbers and permuting them
Thoughts about measurable functions
Can you explain the “Axiom of choice” in simple terms?
Splitting of the tangent bundle of a vector bundle

I’m trying to find the expansion for the roots of this equation. I’ve found one root as $x\sim 1+\epsilon $. Now considering the dominant balance I want to rescale so that

$\epsilon x^3\sim O(x) \Rightarrow x=O(1/\sqrt\epsilon )$

Setting $x=y(1/\sqrt\epsilon )$ where $y=O(1)$ I get the new equation

$$y^3-y+\sqrt\epsilon=0$$

Now I want to substitute in $y\sim y_0+\epsilon y_1+\epsilon ^2 y_2+…$ and equate orders of $\epsilon$, but I’m not sure how to deal with the $\sqrt\epsilon$

Help is much appreicated

- A simple proof that $\bigl(1+\frac1n\bigr)^n\leq3-\frac1n$?
- Filling the gap in knowledge of algebra
- Alternative ways to write $ k \binom{n}{k} $
- Why is the derivative of sine the cosine in radians but not in degrees?
- Minimum value of $2^{\sin^2x}+2^{\cos^2x}$
- On Profit and loss

- Rationalize $\left(\sqrt{3x+5}-\sqrt{5x+11} -\sqrt{x+9}\right)^{-1}$
- Value of $ \cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $?
- Upper bound for the strict partition on K summands
- Prove that $a^2+ab+b^2\ge 0$
- Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$
- Beautiful cyclic inequality
- Prove that $x^2<\sin x \tan x$ as $x \to 0$
- Show that ${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$
- A trigonometric equation
- Proving that for reals $a,b,c$, $(a + b + c)^2 \leq 3(a^2+b^2+c^2)$

Why not to write instead

$$y=a(0)+a(1) \sqrt{\epsilon }+a(2) \epsilon+a(3) \epsilon ^{3/2}+a(4) \epsilon ^2+a(5) \epsilon ^{5/2}+a(6) \epsilon ^3+…$$ Replace $y$ in the equation and start equating the coefficient of each power of $\epsilon$ to $0$. Each power will lead to a linear equation of one coefficient at the time.

If I did not make any mistake, you should arrive at $$y=1-\frac{\sqrt{\epsilon

}}{2}-\frac{3 \epsilon }{8}-\frac{\epsilon ^{3/2}}{2}-\frac{105 \epsilon ^2}{128}-\frac{3 \epsilon ^{5/2}}{2}+\frac{1093 \epsilon

^3}{1024}+ …$$

- Prove that the exponential function is differentiable
- Is it always true that $(A_1 \cup A_2) \times (B_1 \cup B_2)=(A_1\times B_1) \cup (A_2 \times B_2)$
- The system of Diophantine equations.
- How to understand the combination formula?
- Does there exist a matrix $P$ such that $P^n=M$ for a special matrix $M$?
- Is the intersection of two f.g. projective submodules f.g.?
- Show that $f(2n)= f(n+1)^2 – f(n-1)^2$
- What does the Axiom of Choice have to do with right inversibility?
- How many number of functions are there?
- Can every curve be subdivided equichordally?
- Non surjectivity of the exponential map to GL(2,R)
- How $x^4$ is strictly convex function?
- The fractional part of $n\log(n)$
- Show that $\int_{\mathbb{R}} F(x) F(dx)=\frac{1}{2}$ if $F$ is a continuous distribution function
- Probability that the equation so formed will have real roots