Question and Answers Forum

AllQuestion and Answers: Page 479

Question Number 168977 Answers: 0 Comments: 0

check that the function u(x,t) = exp{−((n^2 α^2 π^2 )/L^2 )t} sin((nπx)/L) n = 1,2,... satisfy the heat equation heat equation α^2 (∂^2 u/∂x^2 ) = (∂u/∂t), 0 < x < L

$$\mathrm{check}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function} \\ $$$${u}\left({x},{t}\right)\:=\:\mathrm{exp}\left\{−\frac{{n}^{\mathrm{2}} \alpha^{\mathrm{2}} \pi^{\mathrm{2}} }{{L}^{\mathrm{2}} }{t}\right\}\:\mathrm{sin}\frac{{n}\pi{x}}{{L}} \\ $$$${n}\:=\:\mathrm{1},\mathrm{2},...\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{heat}\:\mathrm{equation} \\ $$$$\boldsymbol{\mathrm{heat}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\alpha^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }\:=\:\frac{\partial{u}}{\partial{t}},\:\mathrm{0}\:<\:{x}\:<\:{L} \\ $$

Question Number 168952 Answers: 1 Comments: 0

Question Number 168948 Answers: 1 Comments: 0

Question Number 168946 Answers: 2 Comments: 1

Question Number 168942 Answers: 0 Comments: 2

Question Number 168956 Answers: 4 Comments: 0

solve ⌊ x ⌋ + ⌊ (x/6) ⌋ =(2/3) x

$$ \\ $$$$\:\:\:\:{solve} \\ $$$$\:\:\lfloor\:{x}\:\rfloor\:+\:\lfloor\:\frac{{x}}{\mathrm{6}}\:\rfloor\:=\frac{\mathrm{2}}{\mathrm{3}}\:{x} \\ $$$$ \\ $$

Question Number 168937 Answers: 1 Comments: 0

Determine m & n such that: digit-sum(m^2 )=n_(&_(digit-sum(n^2 )=m) ) ^■ digit-sum(abc..^(−) .)=a+b+c+...

$$ \\ $$$$\underline{\mathcal{D}{etermine}\:{m}\:\&\:{n}\:{such}\:{that}:} \\ $$$$\:\:\underset{\underset{{digit}-{sum}\left({n}^{\mathrm{2}} \right)={m}} {\&}} {\:{digit}-{sum}\left({m}^{\mathrm{2}} \right)={n}} \\ $$$$\:^{\blacksquare} {digit}-{sum}\left(\overline {{abc}..}.\right)={a}+{b}+{c}+...\:\:\:\:\:\:\:\: \\ $$

Question Number 168968 Answers: 0 Comments: 0

Question Number 168966 Answers: 2 Comments: 0

lim_(x→0) ((∫_0 ^( x) (∫_0 ^( u^2 ) tan^(−1) (1+t)dt)dt )/(x−x cos x)) =?

$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\int_{\mathrm{0}} ^{\:{x}} \:\left(\int_{\mathrm{0}} ^{\:{u}^{\mathrm{2}} } \mathrm{tan}^{−\mathrm{1}} \left(\mathrm{1}+{t}\right){dt}\right){dt}\:}{{x}−{x}\:\mathrm{cos}\:{x}}\:=? \\ $$

Question Number 168964 Answers: 2 Comments: 0

lim_(x→0) ∫_0 ^x ((sin 2t)/( (√(4+t^2 )) ∫_0 ^x ((√(1+t))−1)dt)) dt =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{t}}{\:\sqrt{\mathrm{4}+{t}^{\mathrm{2}} }\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\left(\sqrt{\mathrm{1}+{t}}−\mathrm{1}\right){dt}}\:{dt}\:=?\: \\ $$

Question Number 168926 Answers: 2 Comments: 2

Question Number 168921 Answers: 1 Comments: 0

Question Number 168911 Answers: 1 Comments: 1

Question Number 168910 Answers: 0 Comments: 3

Resolve 1) (x−y)ydx−x^2 dy=0 2)(2x−y)dx+(4x−2y+3)dy=0

$${Resolve} \\ $$$$\left.\mathrm{1}\right)\:\left({x}−{y}\right){ydx}−{x}^{\mathrm{2}} {dy}=\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\left(\mathrm{2}{x}−{y}\right){dx}+\left(\mathrm{4}{x}−\mathrm{2}{y}+\mathrm{3}\right){dy}=\mathrm{0} \\ $$

Question Number 168909 Answers: 0 Comments: 1

Resolve (1−x^2 y)dx+(x^2 y−x^3 )dy=0 ; μ=μ(x)

$${Resolve}\: \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} {y}\right){dx}+\left({x}^{\mathrm{2}} {y}−{x}^{\mathrm{3}} \right){dy}=\mathrm{0}\:;\:\mu=\mu\left({x}\right) \\ $$

Question Number 168906 Answers: 1 Comments: 0

Question Number 168905 Answers: 1 Comments: 0

Question Number 168898 Answers: 1 Comments: 0

Question Number 168894 Answers: 1 Comments: 1

Question Number 168885 Answers: 0 Comments: 11

Find all the values of n such that: determinant (((digit-sum(n^3 )=n ))) and show that n has at most 2 digits. ^■ digit-sum(abc..^(−) )=a+b+c+...

$$ \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}: \\ $$$$\:\:\:\:\begin{array}{|c|}{\mathrm{digit}-\mathrm{sum}\left(\mathrm{n}^{\mathrm{3}} \right)=\mathrm{n}\:}\\\hline\end{array} \\ $$$$\mathrm{and} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{n}\:\mathrm{has}\:\mathrm{at}\:\mathrm{most}\:\mathrm{2}\:\mathrm{digits}. \\ $$$$\:\:^{\blacksquare} \mathrm{digit}-\mathrm{sum}\left(\overline {\mathrm{abc}..}\right)=\mathrm{a}+\mathrm{b}+\mathrm{c}+... \\ $$

Question Number 168881 Answers: 1 Comments: 0

sin(π(√(ix))) + sinh(π(√(ix))) = 0 Find all the real value/s of x.

$$\:\mathrm{sin}\left(\pi\sqrt{\mathrm{ix}}\right)\:+\:\mathrm{sinh}\left(\pi\sqrt{\mathrm{ix}}\right)\:=\:\mathrm{0} \\ $$$$\: \\ $$$$\:\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{value}/\mathrm{s}\:\mathrm{of}\:\mathrm{x}. \\ $$

Question Number 168878 Answers: 1 Comments: 3

solve: x^(2019) +x^(2020) +x^(2021) +x^(2022) =4

$${solve}:\:\:{x}^{\mathrm{2019}} +{x}^{\mathrm{2020}} +{x}^{\mathrm{2021}} +{x}^{\mathrm{2022}} =\mathrm{4} \\ $$

Question Number 168874 Answers: 0 Comments: 0

Question Number 168873 Answers: 0 Comments: 0

Question Number 168872 Answers: 0 Comments: 2

E=∫^π _0 [((a^2 σ sin θ)/(2ε(√(a^2 −x^2 −2ax cosθ))))]dθ If a>x show that E = ((a^2 σ)/(εx))

$${E}=\underset{\mathrm{0}} {\int}^{\pi} \left[\frac{{a}^{\mathrm{2}} \sigma\:\mathrm{sin}\:\theta}{\mathrm{2}\epsilon\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} −\mathrm{2}{ax}\:\mathrm{cos}\theta}}\right]{d}\theta \\ $$$$\mathrm{If}\:{a}>{x}\:\mathrm{show}\:\mathrm{that}\:{E}\:=\:\frac{{a}^{\mathrm{2}} \sigma}{\epsilon{x}} \\ $$

Question Number 168860 Answers: 0 Comments: 0

Pg 474 Pg 475 Pg 476 Pg 477 Pg 478 Pg 479 Pg 480 Pg 481 Pg 482 Pg 483

Contact: info@tinkutara.com