Prove Va e Z, Vb e Z, 3|(a² + b² ) – 3|a ^ 3|b where xy means x divides y, i.e., there exists k e Z, xk = y.

Q: find in Euler's way 2x ! y(0.1) de y(0)=1 , h=0.05 y' = y -- y

A: The general form for Euler's method is given by yn=yn-1+hfxn-1,yn-1.

Q: |(7) In general, SUT S (SUT)', but (SU T)' S°UT. Pra

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Q: 1. What should be A for the path independent of (2,4) xy+ ^ dr + 2Ay – x dx + y? fip- (1,2) Moreover…

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Q: Let Un−1(x) be defined for n ≥ 1and define U−1(x) = 0. Show that Tn(x) = Un(x) − xUn−1(x), for n ≥ 0

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Q: Let z 2ev Then: 8z 8z dy sissis

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Q: xfx+yfy=nf  f is a homogeneous of degree n

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Q: 5 (show resulting de) 2xdx - dy=x(xdy - 2ydx) When: x= -3 , y = 1 %3D

A: Differential equation and it's solution

Q: 7 y=tanx show thats y" = -2Siny cosy 1f y=e Snbx show that:

A: Dear Student, according to the Bartleby guidelines, we can answer only first question if multiple…

Q: Use two different methods to calculate dzu/ar if u = /x² + y². = est, and y = 1 + s² cosz. IUse two…

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Q: Show Sn+3 diverges by osing 6n +5 the test for divegence 1こV

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Q: 2² + e4 z lim z 0 22 – e²

A: This is a form of infinity by infinity so we have to apply Lhospital rule that is differentiate both…

Q: 4. If x = cose + isin0 & 1 + V1– a² = na then prove that 1+ acose =(1+ nx)(1+) %3D 2n

A: Given, x= cos θ+i sinθ and na=1+1-a2      To prove:  1+a cosθ=a2n(1+nx)(1+nx)       Let  x=…

Q: 2/3 Determine the length of x=| for 0<ys2. Assume that y is positive.

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Q: o what does it con s conditionally conv 1. onvergence for at is the radius of approximation of e √2,

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Q: 3. From Eq. (1), deduce that U2n-1 2. U2n 2. %D %3D n 2

A: 3

Q: The evaluation of S V1 + tan2x dx is C In(V2) +1 In(vZ+1) C In(vZ-1) O-In(vZ+1)

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Q: What is G = xy ax – x²y ay + 2y az at B(3, –4, 1), expressed in SCS?

A: Given: G→=xy ax-x2y ay+2y az -----------(1) Where, Bx,y,z=B3,-4,1 --------(2)

Q: Obtain the general soluti d²y 1. 7- y = 10sin²x

A: Given: d2ydx2-y=10 sin2(x)

Q: Are these equation separable 1.s2+ds/dt = (s+1)/st 2.dr= b(cosx dr+ rsinxdx) 3.y'=ye(x+y)/(x2+2)

A: The objective of the problem is to determine whether the given equations are separable. A…

Q: of Let f(x,y)=tanh-1(x-y) with x=e" and y= usinh (1). Then the value of Ot (u,t) =(4,In 2) is equal…

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Q: 4. Using the definition of sech x ,x 0 in terms of e , prove that 1+V1-x2 sech x = In ,0<x <1 Hence,…

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Q: If fufe.- fuave diffa Veubable at'xo eI fuave difle Veubable at Xo eI Pi:I R,ists-n prove O…

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Q: 9 Give with an not example of T: W≤X →Y, to XT Ext. .R. boundedag. - keše bes

A: Please find the solution innext step this is an non-example of Tietze extension theorem in Topology.

Q: 3 1-Vy dy 2 2у3

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: In Exercises 5–12, find and sketch the domain for each function

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Q: dy then In z is If y = cosh z

A: Without wasting any time, any further, differentiate the RHS to get your answer.

Q: dz Find if yz – In z = x + y ax

A: To find the partial derivative of z of yz-ln z=x+y with respect x, differentiate yz-ln z=x+y, while…

Q: Prove that v = In [(x-2)^2 + (y - N)^2] is harmonic in every region does not include .the point ( 2,…

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Q: Thene m for Voritly functiom Mean Valne fex) 2. 3X -4 +S the

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Q: 10. Using the substitution u 2x - 1, prove that %3D 3 8 dx J1 (2x-1) 25

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Q: 1. What should be A for the path independent of (2,4) xy + ) 2Ay – x y? (1,2) Moreover evaluate the…

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Q: ) Fnl the Vdume of the Selid enclesed by the curres "about the x-axS. y=e^, y=0 x =©,'x=/

A: To find out the volume of solid of revolution.

Q: Q1 : prove 1 2(1–z) 3(1--z) dydxdz = 1

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Q: Find the Integration of: Ja (3sint + esin t) cost dt In 3 2

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Q: et + 6e3t + 12e2t +9e dt. (d) Evaluate the integral / (e2t +4et +5)2

A: First, simplify the integral. As the degree of the numerator is greater than the denominator, use…

Q: 2 dt 22. Evaluate o v9– 2t² 2 dy 3-y³ 24. Evaluate o v8-y'

A: 22)I=∫0129-2t2dtt=32 udt=32 duI=∫29-9u2 32duI=2 ∫11-u2 duI=2 sin-1uI=2 sin-12 t301I=2 sin-1(23)

Q: 7) [[3embch5y] dvl [jel*5) dy] „In[sec h 5y]

A: Using rule of logarithm &amp; integration formula of sech(x)

Q: What is the net crarop for fW={+5,05 as we qo from x=2 to X=5?

A: The given function  f(x) = x2 + 5

Q: d 1 Prove that: dz (tan¬1 z) = 1+z?

A: Given query is to find the derivative of the expression.

Q: Find fCx) if flo)=0 and the tanGat (k,F CX) has ine af Slore x(x+2) a

A: Given for the function F(x), F(0)=0 and tangent line to F(x) at x,F(x) has slope xx2+23.

Q: Use Simpson's Rule with n = 4 to evaluate [² (t³ + t) dt

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Q: If u(x, y) = 2 tan-1 2 and v= In (x² + y?) , prove that: a2v = 0 дхду dx2

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Q: Find dr. J (1+z²)3/2 - In Vx² + 1 + x| + In |Vx² + 1 + x| - Ja? + 1 +c Vx² + 1 O Option 3 Option 1…

A: Let I= ∫x21+x232dx Let x=tanθ ⇒dx = sec2θdθ Substituting this in I, we get

Q: vsin ax (02+4)4= sin x is the fun ction with y %3D

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Q: Q2-Prove that = va* – x² ify= ()sin () + () va² - x* = va² – x* if y = (;")sin*+ (4) + () Va² - x²

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Q: Qし If xと -l Sx S2 fa)= { 5-x fo) = 2< x S3 - Is fox) Condinuons at x =2 2 and sketch this funclion

A: Hello, Thank you for the question. According to our honor code can answer only one question at a…

Q: d'y on of x

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Q: 16.1. Use an antiderivative to show that for every extending from a point 21 to a point 22 2" dz =…

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