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let give the sequence (y_n ) /y_0 (x)=1 and y_n (x)= 1+ ∫_0 ^x (y_(n−1) (t))^2 dt , let suppose x∈[0,1] prove that (y_n ) is increasing majored by (1/(1−x)) if y=lim_(n→+∞) y_n prove that y is solution of differencial equation y^, =y^2 and y(o)=1.

$${let}\:{give}\:{the}\:{sequence}\:\:\left({y}_{{n}} \right)\:/{y}_{\mathrm{0}} \left({x}\right)=\mathrm{1}\:\:{and} \\ $$$${y}_{{n}} \left({x}\right)=\:\mathrm{1}+\:\int_{\mathrm{0}} ^{{x}} \left({y}_{{n}−\mathrm{1}} \left({t}\right)\right)^{\mathrm{2}} {dt}\:,\:{let}\:{suppose}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{prove} \\ $$$${that}\:\left({y}_{{n}} \right)\:{is}\:{increasing}\:{majored}\:{by}\:\frac{\mathrm{1}}{\mathrm{1}−{x}}\:{if}\:{y}={lim}_{{n}\rightarrow+\infty} {y}_{{n}} \\ $$$${prove}\:{that}\:{y}\:{is}\:{solution}\:{of}\:{differencial}\:{equation} \\ $$$${y}^{,} ={y}^{\mathrm{2}} \:{and}\:{y}\left({o}\right)=\mathrm{1}. \\ $$

Question Number 29036 Answers: 0 Comments: 0

p=2m+1 is a prime number prove that 1) (p−1)!≡ −1[p] 2) (m!)^2 ≡ (−1)^(m+1) [p]

$${p}=\mathrm{2}{m}+\mathrm{1}\:{is}\:{a}\:{prime}\:{number}\:{prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\left({p}−\mathrm{1}\right)!\equiv\:−\mathrm{1}\left[{p}\right] \\ $$$$\left.\mathrm{2}\right)\:\left({m}!\right)^{\mathrm{2}} \equiv\:\left(−\mathrm{1}\right)^{{m}+\mathrm{1}} \:\left[{p}\right] \\ $$

Question Number 29035 Answers: 0 Comments: 0

let give a prime number p>2 and a /D(a,p)=1 and suppose that the equation x^2 ≡ a[p]have a solution1) 1) prove that a^((p−1)/2) ≡ 1 [p] 2)prove that x^2 ≡ −1[p] ⇔ p≡ 1 [4]

$${let}\:{give}\:{a}\:{prime}\:{number}\:{p}>\mathrm{2}\:\:{and}\:{a}\:/{D}\left({a},{p}\right)=\mathrm{1}\:{and}\: \\ $$$$\left.{suppose}\:{that}\:{the}\:{equation}\:{x}^{\mathrm{2}} \equiv\:{a}\left[{p}\right]{have}\:{a}\:{solution}\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:{a}^{\frac{{p}−\mathrm{1}}{\mathrm{2}}} \:\:\:\equiv\:\mathrm{1}\:\left[{p}\right] \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:{x}^{\mathrm{2}} \equiv\:−\mathrm{1}\left[{p}\right]\:\Leftrightarrow\:\:\:{p}\equiv\:\mathrm{1}\:\left[\mathrm{4}\right] \\ $$

Question Number 29032 Answers: 0 Comments: 0

let give A = (((0 1 0)),((0 0 1)) ) (1 0 0 1) find A^3 2) find e^(tA) .

$$ \\ $$$${let}\:{give}\:{A}\:=\:\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\mathrm{0}\right. \\ $$$$\left.\mathrm{1}\right)\:{find}\:{A}^{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{tA}} \:\:. \\ $$

Question Number 29031 Answers: 0 Comments: 0

find all function f ∈C^1 (R^2 ,R) wich verify (∂f/∂x) −(∂f/∂y)=0 ∀(x,y)∈R^2 .

$${find}\:{all}\:{function}\:{f}\:\in{C}^{\mathrm{1}} \left({R}^{\mathrm{2}} ,{R}\right)\:{wich}\:{verify} \\ $$$$\frac{\partial{f}}{\partial{x}}\:−\frac{\partial{f}}{\partial{y}}=\mathrm{0}\:\:\:\forall\left({x},{y}\right)\in{R}^{\mathrm{2}} . \\ $$

Question Number 29030 Answers: 0 Comments: 0

find Σ_(n=0) ^∞ (t^(3n) /((3n)!)) .

$${find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{t}^{\mathrm{3}{n}} }{\left(\mathrm{3}{n}\right)!}\:. \\ $$

Question Number 29029 Answers: 0 Comments: 0

let give A= (((1 −1)),((4 −3)) ) calculate A^n and e^A .