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Question Number 89973 Answers: 1 Comments: 1

x (dy/dx) −y = x^2 tan ((y/x))

$${x}\:\frac{{dy}}{{dx}}\:−{y}\:=\:{x}^{\mathrm{2}} \:\mathrm{tan}\:\left(\frac{{y}}{{x}}\right)\: \\ $$

Question Number 89970 Answers: 0 Comments: 1

xy (dy/dx) = y^2 ((x^3 /(x^2 +1)))

$$\mathrm{xy}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{y}^{\mathrm{2}} \left(\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\right)\: \\ $$

Question Number 89955 Answers: 1 Comments: 2

9^(x+1) ∤28(3^x )+3=0

$$\mathrm{9}^{\mathrm{x}+\mathrm{1}} \nmid\mathrm{28}\left(\mathrm{3}^{\mathrm{x}} \right)+\mathrm{3}=\mathrm{0} \\ $$

Question Number 89953 Answers: 0 Comments: 1

solvethefollowingequation 5^(2x+y) =625and2^(4x∤2y) =(1/6)

$$\mathrm{solvethefollowingequation} \\ $$$$\mathrm{5}^{\mathrm{2x}+\mathrm{y}} =\mathrm{625and2}^{\mathrm{4x}\nmid\mathrm{2y}} =\frac{\mathrm{1}}{\mathrm{6}} \\ $$

Question Number 89951 Answers: 0 Comments: 1

log _2 (sin (x+((5π)/(12)))) + log _2 (sin (x+(π/(12))))=−1

$$\mathrm{log}\:_{\mathrm{2}} \:\left(\mathrm{sin}\:\left({x}+\frac{\mathrm{5}\pi}{\mathrm{12}}\right)\right)\:+\:\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{sin}\:\left({x}+\frac{\pi}{\mathrm{12}}\right)\right)=−\mathrm{1} \\ $$

Question Number 89950 Answers: 0 Comments: 1

Question Number 89946 Answers: 1 Comments: 0

∫ _(−(π/2)) ^(π/2) (dx/(1+e^(sin x) ))

$$\int\underset{−\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\:}}\:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{e}^{\mathrm{sin}\:\mathrm{x}} } \\ $$

Question Number 89958 Answers: 0 Comments: 1

prove that (1+sin x/1+cos 3(1sin x/1+cosec x)=tanx

$${prove}\:{that}\:\left(\mathrm{1}+\mathrm{sin}\:{x}/\mathrm{1}+\mathrm{cos}\:\mathrm{3}\left(\mathrm{1sin}\:{x}/\mathrm{1}+\mathrm{cosec}\:{x}\right)={tanx}\right. \\ $$

Question Number 89956 Answers: 0 Comments: 1

simplifyκgivingκyourκanswerκinκindexκform (√((ac^2 )/(9a^2 c^4 )))

$$\mathrm{simplify}\kappa\mathrm{giving}\kappa\mathrm{your}\kappa\mathrm{answer}\kappa\mathrm{in}\kappa\mathrm{index}\kappa\mathrm{form} \\ $$$$\sqrt{\frac{\mathrm{ac}^{\mathrm{2}} }{\mathrm{9a}^{\mathrm{2}} \mathrm{c}^{\mathrm{4}} }} \\ $$

Question Number 89938 Answers: 0 Comments: 1

If x(x+1) = 1 find (x+1)^3 −(1/((x+1)^3 ))

$$\mathrm{If}\:\mathrm{x}\left(\mathrm{x}+\mathrm{1}\right)\:=\:\mathrm{1}\: \\ $$$$\mathrm{find}\:\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} −\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 89937 Answers: 0 Comments: 1

Prove that for all complex such as ∣z∣<1= Σ_(n=1) ^∞ (z^n /((z^n −1)^2 )) +Σ_(n=1) ^∞ ((nz^n )/(z^n −1)) = 0

$${Prove}\:{that}\:{for}\:{all}\:{complex}\:{such}\:{as}\:\mid{z}\mid<\mathrm{1}= \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{z}^{{n}} }{\left({z}^{{n}} −\mathrm{1}\right)^{\mathrm{2}} }\:+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{nz}^{{n}} }{{z}^{{n}} −\mathrm{1}}\:=\:\mathrm{0}\: \\ $$

Question Number 89936 Answers: 1 Comments: 0

Prove that Σ_(p≥1,q≥1) (1/(pq(p+q−1))) =(π^2 /3)

$${Prove}\:{that}\:\underset{{p}\geqslant\mathrm{1},{q}\geqslant\mathrm{1}} {\sum}\:\:\frac{\mathrm{1}}{{pq}\left({p}+{q}−\mathrm{1}\right)}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{3}}\: \\ $$

Question Number 89934 Answers: 0 Comments: 0

Let x∈]0;1[ Prove that Σ_(n=1) ^∞ (x^n /(1+x^n )) +Σ_(n=1) ^∞ (((−x)^n )/(1−x^n )) = 0

$$\left.{Let}\:{x}\in\right]\mathrm{0};\mathrm{1}\left[\:\:{Prove}\:{that}\right. \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }\:+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{\mathrm{1}−{x}^{{n}} }\:=\:\mathrm{0} \\ $$

Question Number 89928 Answers: 1 Comments: 0

Question Number 89925 Answers: 0 Comments: 0

x^2 (yy′′−y^2 )+xyy′ = y(√(x^2 (y′)^2 +y^2 ))

$${x}^{\mathrm{2}} \left({yy}''−{y}^{\mathrm{2}} \right)+{xyy}'\:=\:{y}\sqrt{{x}^{\mathrm{2}} \left({y}'\right)^{\mathrm{2}} +{y}^{\mathrm{2}} }\: \\ $$

Question Number 89922 Answers: 0 Comments: 2

Question Number 89918 Answers: 0 Comments: 1

∫ ((x tan^(−1) (x))/((1+x^2 )^(3/2) )) dx

$$\int\:\frac{\mathrm{x}\:\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }\:\mathrm{dx}\: \\ $$

Question Number 89913 Answers: 1 Comments: 1

Question Number 89908 Answers: 0 Comments: 6

Solve the differential equstion: (d^2 y/dx^2 ) = ((y_0 − 2y_(−1) + y_(−2) )/h^2 )

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equstion}: \\ $$$$\:\:\:\:\:\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }\:\:\:=\:\:\:\frac{\mathrm{y}_{\mathrm{0}} \:\:−\:\:\mathrm{2y}_{−\mathrm{1}} \:\:+\:\:\mathrm{y}_{−\mathrm{2}} }{\mathrm{h}^{\mathrm{2}} } \\ $$

Question Number 89907 Answers: 1 Comments: 0

If the sum of 4 numbers is between 53 and 57 then the arithmetic mean of the numbers could be one of the following a)11.5 b)12 c)12.5 d)13 e)14

$${If}\:{the}\:{sum}\:{of}\:\mathrm{4}\:{numbers}\:{is}\:{between} \\ $$$$\mathrm{53}\:{and}\:\mathrm{57}\:{then}\:{the}\:{arithmetic}\:{mean}\:{of} \\ $$$${the}\:{numbers}\:{could}\:{be}\:{one}\:{of}\:{the} \\ $$$${following} \\ $$$$ \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left.\mathrm{1}\left..\mathrm{5}\:{b}\right)\mathrm{12}\:{c}\right)\mathrm{12}.\mathrm{5}\:{d}\right)\mathrm{13}\:{e}\right)\mathrm{14} \\ $$

Question Number 89906 Answers: 1 Comments: 0

Show that 2x^7 −4x^4 +4x^2 =6x^6 +3 Has no solution in N

$${Show}\:{that}\: \\ $$$$\mathrm{2}{x}^{\mathrm{7}} −\mathrm{4}{x}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{2}} =\mathrm{6}{x}^{\mathrm{6}} +\mathrm{3} \\ $$$${Has}\:{no}\:{solution}\:{in}\:\mathbb{N} \\ $$

Question Number 89898 Answers: 0 Comments: 3

In a classroom when the students sit 2 per bench, 11 students are left with no sits. And when they sit 3 per bench 7 benches are left empty. Determine the number of students in this classroom.

$${In}\:{a}\:{classroom}\:{when}\:{the}\:{students}\:{sit} \\ $$$$\mathrm{2}\:{per}\:{bench},\:\mathrm{11}\:{students}\:{are}\:{left}\:{with} \\ $$$${no}\:{sits}.\:{And}\:{when}\:{they}\:{sit}\:\mathrm{3}\:{per}\:{bench} \\ $$$$\mathrm{7}\:{benches}\:{are}\:{left}\:{empty}.\:{Determine}\:{the}\:{number} \\ $$$${of}\:{students}\:{in}\:{this}\:{classroom}. \\ $$

Question Number 89896 Answers: 0 Comments: 4

Question Number 89891 Answers: 0 Comments: 1

calculate ∫_(1/e) ^1 ln(x)ln(1+x)dx

$${calculate}\:\int_{\frac{\mathrm{1}}{{e}}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right){dx} \\ $$

Question Number 89882 Answers: 2 Comments: 0

Question Number 89878 Answers: 1 Comments: 1

A four_digit whole number is interesting if the number formed by the leftmost two digits is twice as large as the number formed by the rightmost two digits. (for example 2010 is interesting) 1 find the largest whole number B such that all interesting numbers are divisible by B 2 find the smallest whole number D such that D is divisible by all interesting numbers.

$$\boldsymbol{{A}}\:\boldsymbol{{four\_digit}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{is}}\:\boldsymbol{{interesting}}\:\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}}\:\boldsymbol{{leftmost}}\:\boldsymbol{{two}} \\ $$$$\boldsymbol{{digits}}\:\boldsymbol{{is}}\:\boldsymbol{{twice}}\:\boldsymbol{{as}}\:\boldsymbol{{large}}\:\boldsymbol{{as}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{number}}\:\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{rightmost}}\:\boldsymbol{{two}}\:\boldsymbol{{digits}}. \\ $$$$\left(\boldsymbol{{for}}\:\boldsymbol{{example}}\:\mathrm{2010}\:\boldsymbol{{is}}\:\boldsymbol{{interesting}}\right) \\ $$$$\mathrm{1}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{largest}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\mathbb{B}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:\boldsymbol{{all}}\:\boldsymbol{{interesting}} \\ $$$$\boldsymbol{{numbers}}\:\boldsymbol{{are}}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathbb{B} \\ $$$$\mathrm{2}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{smallest}}\:\boldsymbol{{whole}} \\ $$$$\boldsymbol{{number}}\:\mathbb{D}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:\mathbb{D}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\boldsymbol{{all}}\:\boldsymbol{{interesting}} \\ $$$$\boldsymbol{{numbers}}. \\ $$

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