Q.No 05: Find the expressions for the spherical harmonics Y30 (0, 9) and Y3,+1(0, 9). Y30 (0, 9) = /7/16x (5 cos 0– 3 cos 0), Y3,41 (0, 9) = F/21/64x sin 0(5 cos 0–1)e*io, in terms of the Cartesian coordinates x, y,z.

Q: 27. Inside the circle r = 3a cos 0 and outside the cardioidr = a(1+cos 60), %3D %3D a > 0

A: Find the area of the region inside the circle and outside the cardioid, where the equation of circle…

Q: A particle moves in the xY plane so that at any time 0 <t < 4, x = 2 cos t and y =t² + 3 sin t + 5…

A:

Q: Let ŕ ). ) - (0**) k (4 Sin 2+) + (4 cos 2 t, be the Position oF Perticle space in at time. Finc the…

A: The velocity is dr/dt and the acceleration is d2r/dt2

Q: When 9-x-y (x²+ y² +z²)% dzdydx the t 2n n 3 to spherical coordinates becomes the form !! p…

A: Solve the triple integral.

Q: O(y-+ rycota)dr + zdy O (coszcosy– cotz)dr sinrsinydy 0 O ydr + (3z -g')dy = 0

A: A differential equation is called linear if 1) every dependent variable and derivative involved…

Q: Q3//A:- Verify that dV = p² sin p dp d0 do when using spherical coordinates.

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

Q: D Eolm the secend order diffefential equation using the parametric delinitien of y=asine X-a Cos e…

A:

Q: 4. Find the length of the curve having a parametric equation of x = a cos 3 6, y = a sin 2 0 from e…

A:

Q: 4. Calculate leme/) at (z,y) = (a, 4). sin(z/g) ey2/2 A ev2/2 B 4

A: Here we have to calculate ∂∂xesin(x/y)=? at (x,y)=(π,4)

Q: III. Find the parametric equations of normal line to the surface given by f(x, y) = tan 2 V + y? at…

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Q: (b) Given that cos (t) Vos(21) sin' (t) Jos(2)" |cos(2t) dy Find for the parametric equation given…

A:

Q: dy = cos 20. de dx Given that r= cos 0 ; = - sin 20 and de The slope of the tangent line to the…

A: Given that: r=cosθdxdθ=-sin2θdydθ=cos2θ

Q: 8. Find the positive constant a(> 0) so that the parametric equation: x = at, y = -t? is a solution…

A:

Q: 2. By changing to cylindrical coordinates, evaluate 2 2π Stt. [²* * cos p² sin³ o dp do dô 0

A: To Evaluate: The integral ∫02π∫0π/4∫02/cosϕρ4sin3ϕ dρdϕdθ by converting it to cylindrical…

Q: Given parametric equations x = 2 cos 0 and y = sin 0. d²y Then dx2 in the point (V2,) is

A: Given parametric equation is  x=2cosθ and y=sinθ

Q: dy h,p For problems 1-3 compute dx for the given set of parametric equations. and dx? 1. x = 7t2 –…

A: 1) Given that x=7t2-9t, y=t6+2t2We have to find the dydxand d2ydx2First we find…

Q: 3. Consider the parametric equations x = 1+ cos(t), y = -2+ sin(t), 0sts2M. 3.1. Find an equation…

A:

Q: 8. Find the positive constant a(> 0) so that the parametric x = at, y = -t? is a solution to y =…

A:

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: Parametric equations for a line:

Q: 3. Consider the parametric equations x = 1+ cos(t), y = -2+ sin(t), 0S <2M. 3.1. Find an equation…

A:

Q: 1. For the parametric equations: x = 4t3, y= 2 +60t – 8t?, find: a) All values of t at which the…

A:

Q: x = a(0 – sin 0), y = a(1 – cos 0). %3D %3D urve followed by the particle as a function of 0.

A: The given parametric equations are, x=a(θ-sinθ)...............(1) y=a(1-cosθ) .............(2)

Q: The position of a particle moving in the xy-plane is given by the parametric equations x(t) =…

A:

Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t = 2n. %3D 2. Find the length…

A: As we know that, The expression for the length of vector-valued curve is: L=∫abx't2+y't2dt   1).…

Q: Transform C = cos p a, - sin p a, + cos p sin p az into spherical coordinates and then at P (2, T/4,…

A:

Q: using

A:

Q: Find an equation y = f(x) for the parametric curve c(0) = (cos(0), 3 cos(0) + 8 sin²(0)) and compute…

A: The objective is to find y(x) and it's it's derivative .

Q: 2y )dx - (x +3 cos y)dy around the parallelogram havi - 6

A:

Q: 4. Solve for the critical points of the given parametric equation: x = t³ – 1; y = t² + t

A:

Q: dx dy = Cos 20. do Given that r= cos 0 ; = - sin 20 and d0 The slope of the tangent line to the…

A: Slopes of tangent is dy/dx . Finding dy/dx at theta = 0.

Q: Given that x(t) = log (3t) and y(t) = t² + 3, which of the following gives a Cartesian form of the…

A:

Q: Q6 Evaluate S[(2xy – y? cos x)dx + (1– 2y sin x + 3x?y?)dy] where C is the arc of the parabola 2x =…

A:

Q: Which of the following are parametric equations for the ellilpse glven by 9. + y? = 1? Choose all…

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Q: 4) Find the critical points of the parametric equations x = 5 – In t y = t + Int

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Q: 1 Find the area of the surface generated by revolving the curve x = cos (3t), y 3+sin (3t) on 0sts…

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Q: Question 9

A:

Q: Find the length of the curve. r(t) = cos(3t) i + sin(3t)j + 3 In(cos(t)) k, 0 <t< n/4 %3D

A: Find the length of the curve:…

Q: dx Given that r cos 0; dy - sin 20 and d0 d0 cos 20. The slope of the tangent line to the polar…

A: We will solve the following.

Q: 2- Find the solution on the exterior of a sphere of radius a 1 or (r F) + Sin e 0 (sin 0) = 3u, u…

A: Given: ∂∂rr2∂u∂r+1sinθ∂∂θsinθ∂u∂θ=3u⇒r2∂2u∂r2+2r∂u∂r+1sinθsinθ∂2u∂θ2+cosθ∂u∂θ=3u…

Q: Q6 Evaluate Sel(2xy' - y? cos x)dx + (1 – 2y sin x + 3x?y)dy] where Cis the arc of the parabola [12]…

A: Evaluate ∫c[(2xy3-y2cosx)dx-(1-2ysinx+3x2y2)dy] where c is the arc of parabola 2x=py2from (0,0)to…

Q: 7e cos z Integrate counterclockwise around the circle |z| = 8. (z – 3)² - 7e cos z dz 2 c (z – 5)

A: By using  Cauchy integral formula we can solve this question.

Q: 8. Solve for the critical points of the parametric equations x = t - 1,y = t² +t

A: To find the critical points of parametric curve x(t)=t2-1, y(t)=t2+t

Q: 2. Given the parametric equations x= 2t+ sin 2t and y=1-cos 21, where t is a parameter, dx 2)

A:

Q: 4. Consider the circular helix given by 7(t) = (3 cos t, 3 sin t,4t). Give an equation for the…

A:

Q: Q2: Find parametric equations of the line tangent to the graph of r(1) at the point where t= to.…

A:

Q: 1-For spherical polar coordinate system Zr Zr (2² exp(- 2a 1 Z. -) sin OCosø 42 `a а Jy„V„dV = 8_ =?…

A: For spherical polar co-ordinate system ψpx=142Za32exp-Zr2aZrasinθcosϕTo find:∫ψpxψpxdV=δxx

Q: 26 Given the parametric equations x=4cos? r and y = 5sint. Find dy when t= dx

A:

Q: By about how much will h(x, y, z) = cos (paixy) + xz2 change if the point P(x, y, z) moves from…

A: The given function is,

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.