# Questions tagged [substitution]

Questions that involve a replacement of variable(s) in an expression or a formula.

1,107 questions
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### Floquet substitution and matrix exponentials

Given $$\dot{y} = A(t)y,\quad A(t)= \begin{pmatrix} 1 + \frac{\cos t}{2 + \sin t} & 0 \\ 1 & -1 \end{pmatrix}, \quad y\in\mathbb{R}^2,$$ I have ...
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### Substitution for the limit of $\lim_{n \to \infty}\frac{f(a^n,(ka)^n)}{g\left(a^n,(ka)^n\right)}$

Let us assume that $-1<a<1$. If want to calculate the following limit $$\lim_{n \to \infty}\frac{f(a^n)}{g(a^n)},$$ for simplification, I use the fact that $a^n \to 0$ when $n \to \infty$ and I ...
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### A nice IMO 1983 inequality from a stronger inequality

If you are interested in IMO $1983$ please see: $$3[a^2b(a-b)+b^2c(b-c)+c^2a(c-a)]\geqq b(a+b-c)(a-c)(c-b),$$ where $a,b,c$ are three side-lengths of a triangle. If $c≠{\rm mid}\{a,b,c\}$, the ...
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### Substitution in a parabolic equation

I've been given the parabolic equation: $3 \frac{∂^2u}{∂x^2} + 6\frac{∂^2u}{∂x∂y} + 3 \frac{∂^2u}{∂y^2} - \frac{∂u}{∂x} - 4\frac{∂u}{∂y} + u = 0$ The questions ask you to find the characteristic ...