2 Let C: z-2|=1 and I = dz. Then C z2-4

Q: dx , if we let u =V 25 – x² , then udu = - xdx. (A True B False

A: Given query is to find statement is true or false.

Q: 4. If u = xy + y²z³, where x = rset, y = rs²e when r = 2, s = 1,t = 0.

A: We have to find solution

Q: If f(x, y) = xe2y + 6xy, then is dydx given by: e2y + 6 등 2e2y + 6 2 O None of these O 2xe2y + 6…

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Q: (1) Let C: r(t) = t³i+t7 for 0 ≤ t ≤ 2. Evaluate y³ ds.

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Q: Evaluate h(2), where h = (gon. s (2) = –ig(2) =?+1 1 f (z) . h(2)

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Q: Let C: |z-il=3 and I = $. , dz. Then e 3z z2 |= 6ti |=0 |= 8Ti |= 4Tti None of these

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Q: (b) If V'u =0 for x2 0 and if u f(y) on x=0, Show that ,y): +(y-5)

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Q: Let C: |z-il=9. Then ſ. 1-z -dz = z(z+1) а) 0 b) 2ni c) -2ni d) 4ni e) — 4лі f) 2п а. b. С. d. е. O…

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Q: (c) Is ergodic (d) Is wide sense stationary

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Q: Let C z-1=3 and1 = §, = dz. Then

A: The solution is given as

Q: 1 Evaluate dz, where (i) C is z+ = 2 .2 +1 C

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Q: Let z = a?y- ay?, x = e-24, y = e*, then which of the following is at O a. 2e-2t – 2e2t O b. 2e2t –…

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Q: Let D-{tx.y.2) : 2+y?s2s 1, z2 o}. Evaluate z dv.

A:

Q: 4. Let c+ d -c a+b . а - с с-d (e+ 2d 20 +1) = (d c a). %3D for some a, b, c, d ER. Then a + b+c+d%D…

A: To find  The value of a+b+c+d

Q: Let A = {x E Z: 8x} and let B={rEZ:12lx}. Then * ACB ВСА None of these A=B

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Q: dz If z(x, y) = x²y, æ = 4, and y = u³ v², find du

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Q: 1: Let w = eVE, where x = 5s2 – 4,2, y = 3r?s?, and : = Find the value of when s = 1 and r = 2.

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Q: How do I prove this?

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Q: Let D={(x,y,z) : x2+y2+z?s 49, z2 0}. Evaluate z dv.

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Q: f(z) =z2 is discontinuous in C

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Q: dw и and if w = x³ + In (²) dv E) + y, x = u² –" + 5 and Find - ди y = 3v + 5u and point…

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Q: Let z(x, y) = xe* + sin xy + y - In 2 then a a) -In 2-2 b) In 2-2 2 c) 2 (²-1) d) - 2 (²+1)

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Q: 3. Show that S4 has the presentation (x, y|x* = y² = (xy) = 1). %3D

A: To show: S4 has the presentation x,y| x4=y2=(xy)3=1

Q: Let h(t) = 3t3.2 7t-3.2 %3D | Then h'(t) is h'(1) is s! (7),.4 and h''(1) is

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Q: 12. Let f(z,y) = < then f(1,2) = (b) 글 (d) 금

A: Given :fx, y=x-yx+y-1

Q: /| (9x + y – 2z) dS. Is Evaluate S: z = x + 0 < x< 4, 0 <y< 7

A: Given: ∫∫S9x+y-2zdSS:z=x+y2, 0≤x≤4, 0≤y≤7

Q: Let C: z-i=9. Then ſ. 1-z dz z(z+1) a) 0 b) 2ni с) —2лі d) 4ni e) — 4пі f) 2n

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Q: m+n If x"y" = (x+y)*** m, n d'y prove that = 0 .2 dx?

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Q: Find and d?z and prove that 2 0²z IF 2 = f(xt ay) + F CX-ay) %3D

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Q: (3) = 2 u + U, =D2 at ucx, t) fipdy ing the following initial condition : - -3X U(x,0) = 6e

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Q: 1 Let D = (x, y) : 9 < x² + y² < 25, –y < x < V3 /| 4æydA Evaluate

A: Consider the region D as shown in the figure below: Here, the objective is to find the double…

Q: ở =, then If f(x, y, z) = In(2 + x² – 2y² – 3z), Duf(2, –1, 1) = A) % 11 V3 В) 11 C) D) - E) OC O D…

A: The solution is given as

Q: Q: If u =f(x² +2yz, y² + 2=x)Prove that (y-ax

A: Step by step compute the derivative with respect to each variable x,y or z. Then substitute it in…

Q: If u = is X then uor y OA. 33 OB. - 4 Vyx y a - 8 Vy x3 OD. 4 Vy x3

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Q: дz (B)If Z = f(x, y), x= s² - t², y = 2st. Prove that s მs 2 (s² + t²) of əx

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Q: B 2. 7.2.Preve that CD= 24 SnR Vzt2 cosx

A: Use trigonometric ratio to solve

Q: df Let f(x,y,z)=xy+z?, x =s², y =2rs, z =r?. Find when S =1 and r=5. ds

A: Given that f(x,y,z)=xy+z^2 ,x=s^2 , y=2rs ,z=r^2 We have to find ∂f∂swhen s=1 and r=5

Q: Q5/ If w = f(x,y),x = rcos0,y = rsin0 show that: af af 5² + 1 dw Əw. -)² + ar r2 a0 дө

A: The chain rule can be used to calculate the derivative of the complex functions. The function which…

Q: 5) Let X be a r.v. from F(n, m), Let Y=- then F(m,n). Y a) True b) False

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: show: o f(2) dz f(2)dz (Z-Zo) (C) (2-20)2 (C)

A:

Q: 6. Given M = {(x, y), 1 < x² + y² < 4} C R². (a) Determine dM and ÔM. (b) Is M connected? Pathwise…

A: Given that M=x,y,1≤x2+y2&lt;4⊂R2 (a) The objective is to find dM and ∂M dM=x,y|x2+y2=1 and,…

Q: Q/Find af/ax and af/ay xy Iny

A: The given function is fx,y=exylny To find: ∂f∂x and ∂f∂y

Q: 4 Ruret MC w co , , d, 7 the y a Fid e co f de

A:

Q: Let C: |z – i| = 3 and I = $. dz. Then e 6z z2

A: By using Cauchy integral's formula for nth derivative.

Q: Let D={ (x, : x2+y2+z²< 1600, z2 Evaluate z dv.

A:

Q: Let ř =xi+y+ zk, and let u= ||F|I. 2-2u = 3ue. Show thet Dflu)= 6- 6u)e" r.

A: Given, fu=3u2e-2u. We have to show that ∇fu=6-6ue-2ur→ where r→=xi^+yj^+zj^.

Q: Let f(x, y) = x² + y2 + 2x + 4 and P,(-1,0) then Vfle, is i – j. True False

A:

Q: If f(x, y, z) = In(2 + x² – 2y² – 3z), 7 =, then Duf(2, –1, 1) = - A) * V3 B) 11 C) V3 D) E)

A: The given function is:    f(x,y,z)=ln2+x2-2y2-3z

Q: If F(x,y, z) = (5x – y + az)î + (bx + 2y + 5z)ĵ + (2x + cy – z)k is irrotational then find the value…

A: Explanation of the answer is as follows

Q: Evaluate z cosz dz where C is (1)|z – 1| = 1 (i)피 = 2

A: Answer

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.