Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [substitution]

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

0
votes
1answer
24 views

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 ...
0
votes
1answer
12 views

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 ...
0
votes
2answers
61 views

Find the minimum value of $abc$.

$a$, $b$, $c$ are three positives and $m$, $n$, $p$, $x$, $y$, $z$ are positive parameters . Find the minimum value of $abc$ such that the following inequation is correct. $$\large \dfrac{x}{x + ma} + ...
1
vote
1answer
25 views

Calculate the derivative of underintegral function and find the integral

I have the integral: $$ \Phi(\alpha)=\int\limits_{1}^{2\alpha}\frac{\cos(2\alpha x^3)}{x}\,\mathrm{d}x \tag{1}$$ Have to find $\Phi'(\alpha)$ and calculate the integral then. do I understand ...
0
votes
2answers
58 views

Find the relation between $m$ and $n$ such that the following equation has four roots. [on hold]

Find the relation between $m$ and $n$ such that the following equation has four roots with $m > 0$. $$x^2 + \left(\dfrac{mx}{m + x}\right)^2 = n$$ Well, I know what the answer is. I just want to ...
0
votes
3answers
82 views

$a,b,c>0$ and $abc=1$; prove $\sum_{cyc}\frac1{(b+1)^2}+\frac1{a+b+c+1}\ge1$ [duplicate]

Let $a$, $b$ and $c$ be three positives such that $abc=1$. Prove that $$\sum_{cyc}\frac1{(b+1)^2}+\frac1{a+b+c+1}\ge1$$ Here's what have I done that is completely incorrect. Let $a + b + c + 1 = x$. ...
1
vote
1answer
29 views

Recursive Definition of Normal Form with explicit substitution

Context I assume a simply typed lambda calculus, probably written with de-bruijn indexes. With $\to_\beta$ I denote the $\beta$-reduction as a relation. Also, my question eventually will use this $\...
1
vote
3answers
117 views

Find the maximum of the expression: $ \frac{(а-b)^2(b-с)^2(с-а)^2}{ ((а-b)^2+(b-с)^2+(с-а)^2)^3}$ [on hold]

Find the maximum of the expression : $$ \frac{(а-b)^2(b-с)^2(с-а)^2} { ((а-b)^2+(b-с)^2+(с-а)^2)^3}$$ At first I tried to solve this problem with the help of inequality A.M-G.M, I know the answer ...
2
votes
1answer
98 views

Prove the inequality $\sum x+6\ge 2(\sum\sqrt{xy}) $

Let $x;y;z\in R^+$ such that $x+y+z+2=xyz$. Prove that $$x+y+z+6\ge 2(\sqrt{xy}+\sqrt{yz}+\sqrt{xz}) $$ This inequality is not homogeneous and look at the condition i thought that i would ...
0
votes
1answer
27 views

u Substitution in double integral

$\mu_{Z}^{} = E(Z = \sqrt{(X^{2} + Y^{2})} = \int_{0}^{1}\int_{0}^{1}(x^{2}+y^{2})^{\frac{1}{2}}(4xy)dxdy$ Pulling out constant 4y Step 1: $\mu_{Z}^{} = \int_{0}^{1}4y\int_{0}^{1}(x^{2}+y^{2})^{\frac{...
0
votes
4answers
31 views

Integral $\int \frac{1}{\sqrt{1-2x-x^2}} \, dx$

I have simple integral that must be solved by substitution. $$\int \frac{1}{\sqrt{1-2x-x^2}} \, dx = \int \frac{1}{\sqrt{2-(x+1)^2}} \, dx $$ After substitution $$u = x+1 $$ we get $$\int \frac{1}{\...
2
votes
1answer
54 views

$f\left ( a,\,b,\,c \right )\leqq f\left ( a,\,1,\,c \right )$ with $abc= 1$ and $a,\,b,\,c> 0$

Give $3$ positve numbers $a,\,b,\,c$ such that $abc= 1$ , prove: $$f\left ( a,\,b,\,c \right )= \frac{3\,a+ 2\,b}{\sqrt{5\,a^{\,2}- ab+ b^{\,2}}}+ \frac{3\,b+ 2\,c}{\sqrt{5\,b^{\,2}- bc+ c^{\,2}}}\...
1
vote
1answer
42 views

Does multiplying a function $g(x)$ by $\frac{f(x)}{f(x)}$ preserve the domain?

I feel unsure how to phrase this question so I'm going to start with an example of what I'm wondering about. Say I want to calculate $\int \frac{\sqrt{x-2}}{x-1}dx$. I have been taught one can do so ...
0
votes
1answer
33 views

Making the condition homogeneous

I saw it on AoPS, very great: https://artofproblemsolving.com/community/c6h1274759p6726915 JunBo-Yang used his own substitution for the condition: $a^{\,2}+ b^{\,2}+ c^{\,2}+ 3\,abc= 6$ , then: $$a= \...
7
votes
1answer
141 views

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 ...
0
votes
4answers
115 views

Can you add an extra $e^x$ when integrating?

So I've been given this problem to solve (pretend it's a fraction or click the link to see the question please) $$\int \frac{-26e^x-144}{e^{2x} + 13e^x + 36}dx$$ and I got this far: $$-2\int\frac{...
2
votes
0answers
44 views

Formalizing the notation and definition for the operation of substitution

Substitution is easily one of the most important operations in all of mathematics, to the point that almost everything beyond basic arithmetic is impossible without it. Despite this, I rarely see ...
0
votes
0answers
41 views

Fourier partial sums of Sawtooth wave are not equal its convolution with the Dirichlet kernel!

Let $f$ be the $2\pi$-periodic function relating \begin{equation} f(x) = \frac{\pi-x}{2} \end{equation} on $(0, 2\pi)$. The coefficients of its Fourier series are easily calculated [see (*), ...
3
votes
1answer
105 views

Inequality proof (strange)

Given $a^2 +b^2 +c^2 +d^2 =1$ where $a,b,c,d$ are positive real numbers, prove that $a+b+c+d-1 \geq 16abcd$ How can I prove the inequality ? My attempts: By Cauchy-Schwarz : $ (a+b+c+d)^{2} \leq (...
2
votes
1answer
49 views

Prove that in every triangle the inequality $a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $ takes place

Prove that in every triangle the inequality $$a^3r_a + b^3r_b + c^3r_c \ge 8S(2R-r)^2 $$ takes place, with the usual notations ($a,b,c$ lengths of sides, $r_a, r_b, r_c$ radii of coresponding ...
1
vote
2answers
119 views

$\frac{a}{a^a+1}+\frac{b}{b^b+1}+\frac{c}{c^c+1}\leq \frac{3}{2}$ with $abc=1$

Let $a,b,c>0$ such that $abc=1$ then we have : $$\frac{a}{a^a+1}+\frac{b}{b^b+1}+\frac{c}{c^c+1}\leq \frac{3}{2}$$ My try : The original inequality is equivalent to : $$a(b^b+1)(c^c+1)+b(a^a+1)(...
0
votes
0answers
41 views

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 ...
1
vote
1answer
34 views

Integration by Substitution in $\int_0^{\infty}x^r\frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}[\sin(2\pi\log x)]dx$

In Casella and Berger (2002) I found an example for non-unique moments (example 2.3.10 on page 64). They are providing the following 2 pdfs: $f_1(x) = \frac 1{\sqrt{2\pi}x}e^{-(\log x)^2/2}$, where $...
1
vote
2answers
54 views

3-variable symmetric inequality

Given $a,b,c>0$ satisfying $a^2+b^2+c^2=3$. Prove that $$2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)+3(a+b+c)\geq 15.$$ I've tried to use the inequality $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}...
2
votes
1answer
30 views

Integration by substitution in different dimensions

I want to solve a specific integral, by using substitution. As it is too specific to describe my situation and probably also not of general interest, let me give a toy example. Let $\overline{\Omega} ...
2
votes
2answers
65 views

Find maximum value by using AM-GM inequality

I have a problem: Find the maximum value of $P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$ with $x,y,z>0$. Is there anyway to ...
0
votes
4answers
53 views

Show that $a_n = 5a_{n-1} - 6a_{n-2}$ for all integers $n$ with $n \geq 2$

Let $a_n = 2^n + 5 \cdot3^n$. I'm having a hard time understanding the concept of iteration that needs to be applied to prove this. I understand I can substitute the following: $$a_n = 5(2^{n-1}+ 5 ...
2
votes
2answers
58 views

Inequality with a+b+c=1 and $18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3)$

Let $a,b,c$ be reals with $a+b+c=1$. Show that : $$18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3).$$ I have tried to something like this: $$18a^4-24a^3+6a^2-12a+12\geq 0$$ $$18b^4-24b^3+6b^2-12b+...
2
votes
2answers
111 views

show this inequality $\sum\frac{x}{2+xy}\ge\frac{1}{2}$

Let $x,y,z\ge 0$ such that $$x+y^2+z^3=1.$$ Show that $$\dfrac{x}{2+xy}+\dfrac{y}{2+yz}+\dfrac{z}{2+zx}\ge\dfrac{1}{2}$$ I try do $$\sum_{cyc}\dfrac{x}{2+xy}=\sum_{cyc}\dfrac{x^2}{2x+x^2y}\ge\...
3
votes
0answers
46 views

Integration by inverse substitution - some necessary condition(s)

This image: from this question says that inverse substitution requires the substitute to be one-to-one. But whether $g$ is one-to-one or not, I thought one could write $$ \int f(x)dx = \int f(g(t))...
4
votes
2answers
70 views

If $a$, $b$ and $c$ are sides of a triangle, then prove that $a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c)$ $\leqslant$ $3abc$

Let $a$, $b$ and $c$ be the sides of a triangle. Prove that $$a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c) \leqslant 3abc$$ SOURCE: BANGLADESH MATH OLYMPIAD (Preparatory Question.) I am ...
1
vote
1answer
41 views

Substitution Theorem on “A Concise Introduction to Mathematical Logic” by W. Rautenberg

I'm trying to understand the Substitution Theorem (Theorem 3.5, page 71) in "A Concise Introduction to Mathematical Logic" by W. Rautenberg. At some point Rautenberg states: "The reader should recall ...
1
vote
1answer
84 views

Showing an inequality using Cauchy-Schwarz

I managed to solve the following inequality using AM-GM: $$ \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)} \geq \frac{3}{4} $$ provided that $a,b,c >0$ and $abc=1$. However it was ...
4
votes
3answers
80 views

Determine a valid substitution for a differential equation

I am trying to determine whether any of the $v(x)$ substitutions that I'm given are are possible to make the equation first order linear in terms of $v$. $$y' = \frac{y}{x^2} + x^3y^3$$ The given ...
8
votes
2answers
100 views

If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$?

It is easily seen that if $x,y\in[0,\pi/2)$ satisfy $\tan(x)\tan(y)=1$, then $$\cos(x)+\cos(y)\le\sqrt 2$$ A much more delicate fact is that if $\tan(x)\tan(y)\tan(z)=1$ (while $0\le x,y,z<\pi/2$),...
0
votes
2answers
62 views

Integrate $x^2 \sqrt[]{3+5x^2}dx$ (preferably) using substitution

Today we went over solving integrals with tables. My task is to integrate the following: $\int x^2 \sqrt[]{3+5x^2}dx$ In the back of the book, I am provided with over $100$ integrals. I believe ...
1
vote
1answer
40 views

Some interesting systems of equations [closed]

1, Solve the system of equations:$\left\{\begin{matrix} x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z \\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2} \end{matrix}\right.$ 2,Solve the system of ...
2
votes
1answer
41 views

Finding solution of recurrence relations $T(n) = 2T(n/3)+n$

Evaluate: $$T(n)=2T(n/3)+n$$ $$T(n/3)=2T(n/9)+n/3$$ $$T(n/9)=2T(n/27)+n/9$$ Substitute the following result: $$T(n)=2(2(2T(n/27)+n/9)+n/3)+n$$ $$T(k)=2^k T(n/3^k)+n/3^{k-1}+n/3^{k-2}+....+n/3^...
0
votes
1answer
24 views

Inequality of cyclic expression

For $a,b,c,d$ positive given that $abcd=1.$ Look at cyclic expression (i.e. rotating values in order that of a-b-c-d-a doesn't affect the result of equation) $E$ s.t. $E=\sum_{abcd}\frac{a}{da+a+1}$ ...
3
votes
1answer
76 views

$\frac{1}{3}\sum_{cyc}\frac{1}{\sqrt{1+a}}\ge\frac{1}{\sqrt{1+\sqrt[3]{abc}}}$ if $\;a, b, c\;$ are positive reals s.t. $abc\ge2^9$

$$\frac{1}{3}\sum_{cyc}\frac{1}{\sqrt{1+a}}\ge\frac{1}{\sqrt{1+\sqrt[3]{abc}}}$$ if it is given that $\;a, b, c\;$ are positive reals s.t. $abc\ge2^9$. I have tried (many) dead-end solutions. ...
0
votes
2answers
22 views

Substitution in deduction - Propositional logic

Consider the following deduction step: $$ ((\neg\neg\beta\to\neg\neg\alpha)\to((\neg\neg\beta\to\neg\alpha)\to\neg\beta) $$ $$((\beta\to\alpha)\to((\beta\to\neg\alpha)\to\neg\beta) $$ I have applied $\...
0
votes
1answer
23 views

How to understand the definition of an inverse substitution when finding the primitive function?

From textbook: Inverse substitutions: Let $f$ be a function defined on an interval $I$. Let $g$ be a function from an interval $J$ into an interval $I$ which is differentiable on $J$, and let $h$ ...
4
votes
3answers
276 views

$\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$ if $a+b+c=ab+bc+ca$?

For $a,b,c>0 $ and $a+b+c=ab+bc+ca$ . Prove or disprove that : $\sqrt{24ab+25}+\sqrt{24bc+25}+\sqrt{24ca+25}\geq 21$ I checked in very many cases. Example :$c=1, a=2,b=\frac{1}{2}...$ then it’s ...
2
votes
2answers
69 views

How can $f(x,y)$ be written as a function of $g\left(\frac{y}{x}\right)$?

If $f(tx,ty)=f(x,y)$, how does it imply that: $$f(x,y)=g\left(\dfrac {y}{x}\right)$$ Yesterday I asked a question on ordinary differential equations and got an answer. The answerer used the above ...
2
votes
1answer
21 views

Substitution in complex integral and the Argument Principle.

Let's say $C$ is a simple closed curve in the complex plane and $f(z)$ is holomorphic and doesn't vanish on $C$. According to wikipedia, one can make the following change of variables: $$\omega = f\...
1
vote
1answer
61 views

Elementary differential equations: Please give a proof of the trick

Here is a statement (or theorem) from my book: $\pmb{\text{A neat trick: Turning nonlinear separable equations into linear separable equations}}$ $\mathcal{\text{Knowing when to substitute:}}$ ...
0
votes
1answer
66 views

Find anti-derivative of hairy expression under radical

I'm lost on how to pull the anti-derivative of the expression under the radical. The full integral I'm evaluating is: $$2\pi\int_{16}^{25}(9-{\sqrt{x}})^2{\sqrt{1+\left(\frac{9}{\sqrt{x}}\right)^2}}...
1
vote
1answer
26 views

Integration by substitition: replacing function of x by function of u

I'm having trouble understanding a specific step used to solve the integral below, where instead of replacing $g(x)$ by $u$, it is replaced by a function of $u$: $tan(u)$ $$\int sin(x)\sqrt{1+cos^2(x)...
0
votes
0answers
47 views

Suitable ansatz for Coupled system of PDEs

I have the following three PDEs \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - \...
3
votes
1answer
42 views

integration by substitution of multiple variables

I have an integral \begin{equation} \int_{\mathbb{R}^n}f(\mathbf{B}\mathbf{x})\mathrm{d}\mathbf{x} \end{equation} where $f: \mathbb{R}^m \rightarrow \mathbb{R}$ and $\mathbf{B}\in\mathbb{R}^{m\times ...