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AllQuestion and Answers: Page 1171

Question Number 96600 Answers: 1 Comments: 0

Question Number 96571 Answers: 2 Comments: 0

solve y^(′′) −2y^′ +y =(x+1)^2 e^(−x) with y(0)=−1 ,y^′ (0)=0 ,y^((2)) (0) =1

$$\mathrm{solve}\:\mathrm{y}^{''} −\mathrm{2y}^{'} \:+\mathrm{y}\:=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \:\mathrm{e}^{−\mathrm{x}} \:\mathrm{with}\:\mathrm{y}\left(\mathrm{0}\right)=−\mathrm{1}\:,\mathrm{y}^{'} \left(\mathrm{0}\right)=\mathrm{0}\:,\mathrm{y}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)\:=\mathrm{1} \\ $$

Question Number 96570 Answers: 0 Comments: 0

solve xy^(′′) −(x^3 +1)y^′ +2y =x^2 e^(−x) with y(o)=1 and y^′ (0)=−2

$$\mathrm{solve}\:\:\mathrm{xy}^{''} \:−\left(\mathrm{x}^{\mathrm{3}} +\mathrm{1}\right)\mathrm{y}^{'} \:+\mathrm{2y}\:=\mathrm{x}^{\mathrm{2}} \:\mathrm{e}^{−\mathrm{x}} \:\mathrm{with}\:\:\mathrm{y}\left(\mathrm{o}\right)=\mathrm{1}\:\mathrm{and}\:\mathrm{y}^{'} \left(\mathrm{0}\right)=−\mathrm{2} \\ $$

Question Number 96567 Answers: 1 Comments: 1

Question Number 96558 Answers: 1 Comments: 0

Let m and n be two positive integers satisfy (m/n) = (1/(10×12))+(1/(12×14))+(1/(14×16))+...+(1/(2012×2014)) find the smallest possible value of m+n

$$\mathrm{Let}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}\:\mathrm{be}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{integers}\: \\ $$$$\mathrm{satisfy}\:\frac{\mathrm{m}}{\mathrm{n}}\:=\:\frac{\mathrm{1}}{\mathrm{10}×\mathrm{12}}+\frac{\mathrm{1}}{\mathrm{12}×\mathrm{14}}+\frac{\mathrm{1}}{\mathrm{14}×\mathrm{16}}+...+\frac{\mathrm{1}}{\mathrm{2012}×\mathrm{2014}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{m}+\mathrm{n}\: \\ $$

Question Number 96554 Answers: 1 Comments: 2

∫(dx/(x!))=?

$$\int\frac{\mathrm{dx}}{\mathrm{x}!}=? \\ $$

Question Number 96548 Answers: 0 Comments: 1

Question Number 96543 Answers: 1 Comments: 0

Question Number 96542 Answers: 0 Comments: 6

Question Number 96538 Answers: 1 Comments: 0

Question Number 96530 Answers: 0 Comments: 1

If: (1+m)^3 −3m^3 =2 0≤m≤1 Find: (1+m)^2 −3m^2

$$\boldsymbol{\mathrm{If}}:\:\left(\mathrm{1}+{m}\right)^{\mathrm{3}} −\mathrm{3}{m}^{\mathrm{3}} =\mathrm{2}\:\:\:\:\:\:\:\:\mathrm{0}\leqslant{m}\leqslant\mathrm{1} \\ $$$$\boldsymbol{\mathrm{Find}}:\:\left(\mathrm{1}+{m}\right)^{\mathrm{2}} −\mathrm{3}{m}^{\mathrm{2}} \\ $$

Question Number 96527 Answers: 2 Comments: 1

proof that 1^2 +2^2 +3^2 +....+n^2 =((n(2n+1)(n+1))/6)

$$\mathrm{proof}\:\mathrm{that}\:\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +....+\mathrm{n}^{\mathrm{2}} =\frac{\mathrm{n}\left(\mathrm{2n}+\mathrm{1}\right)\left(\mathrm{n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$

Question Number 96524 Answers: 0 Comments: 1

∫(sinx^(cosx^(sinx) ) −cosx^(sinx^(cosx) ) )dx

$$\int\left(\mathrm{sinx}^{\mathrm{cosx}^{\mathrm{sinx}} } −\mathrm{cosx}^{\mathrm{sinx}^{\mathrm{cosx}} } \right)\mathrm{dx} \\ $$

Question Number 96523 Answers: 0 Comments: 0

Question Number 96514 Answers: 1 Comments: 3

∫_0 ^1 (1/(1+[(1/x)]))dx=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{\mathrm{1}}{\mathrm{1}+\left[\frac{\mathrm{1}}{\boldsymbol{{x}}}\right]}\boldsymbol{{dx}}=? \\ $$

Question Number 96508 Answers: 1 Comments: 0

Question Number 96505 Answers: 2 Comments: 2

((√((8)^(1/(4 )) −(√((√2)+1))))/((√((8)^(1/(4 )) +(√((√2)−1))))−(√((8)^(1/(4 )) −(√((√2)−1)))))) ?

$$\frac{\sqrt{\sqrt[{\mathrm{4}\:\:}]{\mathrm{8}}−\sqrt{\sqrt{\mathrm{2}}+\mathrm{1}}}}{\sqrt{\sqrt[{\mathrm{4}\:\:}]{\mathrm{8}}+\sqrt{\sqrt{\mathrm{2}}−\mathrm{1}}}−\sqrt{\sqrt[{\mathrm{4}\:\:}]{\mathrm{8}}−\sqrt{\sqrt{\mathrm{2}}−\mathrm{1}}}}\:? \\ $$

Question Number 96500 Answers: 2 Comments: 0

If x and y real number satisfy (x+5)^2 +(y−12)^2 =196 , then the minimum value of x^2 +y^2 is

$$\mathrm{If}\:{x}\:{and}\:{y}\:{real}\:{number}\:{satisfy} \\ $$$$\left({x}+\mathrm{5}\right)^{\mathrm{2}} +\left(\mathrm{y}−\mathrm{12}\right)^{\mathrm{2}} =\mathrm{196}\:,\:\mathrm{then}\: \\ $$$$\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{is}\: \\ $$

Question Number 96495 Answers: 2 Comments: 0

calculateI = ∫_0 ^(π/2) ln(cosx +sinx)dx and J =∫_0 ^(π/2) ln(cosx−sinx)dx

$$\mathrm{calculateI}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{cosx}\:+\mathrm{sinx}\right)\mathrm{dx} \\ $$$$\mathrm{and}\:\mathrm{J}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{cosx}−\mathrm{sinx}\right)\mathrm{dx} \\ $$

Question Number 96489 Answers: 1 Comments: 0

Given { ((u_0 =a)),((u_(n+1) =a+(1/2)(1−(1/b^n ))u_n )) :} a>0 b>1 a\ Calculate u_1 , u_2 , and u_3 . b\ Show that the sequence (u_n )_(n∈N) is increasing. c\ Deduce that (u_n )_(n∈N) converges and determine its limit.

$$\mathcal{G}\mathrm{iven}\:\:\begin{cases}{\mathrm{u}_{\mathrm{0}} =\mathrm{a}}\\{\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{a}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{b}^{\mathrm{n}} }\right)\mathrm{u}_{\mathrm{n}} }\end{cases}\:\:\mathrm{a}>\mathrm{0}\:\:\mathrm{b}>\mathrm{1} \\ $$$$\mathrm{a}\backslash\:\:\mathcal{C}\mathrm{alculate}\:\:\mathrm{u}_{\mathrm{1}} ,\:\:\mathrm{u}_{\mathrm{2}} ,\:\:\mathrm{and}\:\:\mathrm{u}_{\mathrm{3}} . \\ $$$$\mathrm{b}\backslash\:\:\mathcal{S}\mathrm{how}\:\:\mathrm{that}\:\:\mathrm{the}\:\:\mathrm{sequence}\:\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\:\mathrm{is}\:\:\mathrm{increasing}. \\ $$$$\mathrm{c}\backslash\:\:\mathcal{D}\mathrm{educe}\:\:\mathrm{that}\:\:\left(\mathrm{u}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\:\mathrm{converges}\:\:\mathrm{and}\:\:\mathrm{determine}\:\:\mathrm{its}\:\:\mathrm{limit}. \\ $$

Question Number 96481 Answers: 0 Comments: 5

∫(1/dx)=?

$$\int\frac{\mathrm{1}}{\mathrm{dx}}=? \\ $$

Question Number 96479 Answers: 1 Comments: 0

show that ∫_1 ^e ((x−xln(x)+1)/(x(x+1)^2 +x ln^2 (x)))dx=arctan((1/(e+1)))

$${show}\:{that} \\ $$$$\int_{\mathrm{1}} ^{{e}} \frac{{x}−{xln}\left({x}\right)+\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)^{\mathrm{2}} +{x}\:{ln}^{\mathrm{2}} \left({x}\right)}{dx}={arctan}\left(\frac{\mathrm{1}}{{e}+\mathrm{1}}\right) \\ $$

Question Number 96467 Answers: 1 Comments: 0

Suppose y = 8 ; (dy/dx) = 4 & (d^2 y/dx^2 ) ∣_(x=1) = −2 . Find the value of (1) ((d(xy))/dx) ∣_(x=1) (1)((d^2 (xy))/dx^2 ) ∣_(x=1)

$$\mathrm{Suppose}\:\mathrm{y}\:=\:\mathrm{8}\:;\:\frac{{dy}}{{dx}}\:=\:\mathrm{4}\:\&\: \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:\mid_{{x}=\mathrm{1}} \:=\:−\mathrm{2}\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\left(\mathrm{1}\right)\:\frac{{d}\left({xy}\right)}{{dx}}\:\mid_{{x}=\mathrm{1}} \\ $$$$\left(\mathrm{1}\right)\frac{{d}^{\mathrm{2}} \left({xy}\right)}{{dx}^{\mathrm{2}} }\:\mid_{{x}=\mathrm{1}} \: \\ $$$$ \\ $$

Question Number 96466 Answers: 0 Comments: 0

Question Number 96460 Answers: 1 Comments: 8

Question Number 96454 Answers: 1 Comments: 0

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