1. Let P(x, y) be the terminal point on the unit circle deter- mined by t. Then sin t = cos t = and tan t =

Q: 1- Find the solution to the vibrating- string problem With Uu = c*uxx , u(0, t) = u(l, t) = 0 3nx…

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Q: Eliminate the parameter t from the following. x = sin t − 5, y = cos t − 4

A: Eliminate the parameter and simplify it.

Q: Let V = eP cos(qr), with p = x In(xyz), q = z cos(Ty) and r = 2 tan-(y). By using the chain rule,…

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Q: 15. Find a) fxedx b) dx using the substitution of x- cos t.

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Q: (c) x(t) = eat cos s wtu(t), a > 0 (d) x(t) = e-af sin w,tu(t), a > 0 %3D

A: Let xt is a function. The Fourier transform of the function xt is denoted by Xiwor Xf and defined by…

Q: 8. If r = sin (ƒ(t)), ƒ(0) = pai/3, and ƒ′(0) = 4, then what is dr/dtat t = 0?

A: Chain Rule d(f(g(x))dx   =   f'(g(x)) . g'(x) We have given r = sin(f(t))        f(0) =  π3  ,…

Q: 19.Show that W[eλt cos(μt), eλt sin(μt)] = μe2λt.

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Q: Eliminate the parameter t from the following. x = cos t − 2, y = sin t + 5

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Q: If y = cos ¹ prove Zyy + z = 0. )Z = (1 + i)(2-1)-¹, find |Z|.

A: Here given y=cos-1(z/x). We prove that zyy+z=0.

Q: 4- if U(x,y) = x2 + 2y and x(r, t) = r sint, y(r,t) = sin?t, au find 1- ar au and 2- at au 1- ar ди…

A: Given that,   U(x,y) = x2+2y   and x(r,t) = rsint,     y(r,t) =sin2t we have to find ∂u∂r,    ∂u∂t…

Q: 1. Prove that the composite trapezoidal rule with h = 2-i, i E N, is exact for ſ cos? (Tx) dr

A: We know that in composite trapezoidal rule, we approximate the integral as follows: ∫abf(x) dx ≈  h…

Q: Compute the Jacobian of : O(r, 0) = (9r cos 0, 4r sin 0)

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Q: If u(t) = (sin(4t), cos(2t), t) and v(t) = (t, cos(2t), sin(4t)), use Formula 4 of this theorem to…

A: Find u(t) . v(t) first then differentiate with respect to “t”.

Q: If y = xsin then y is equal to = X" at x =

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Q: 1. If H(y) = cos y – sin y, Find H (0), H (),H (1T), and also show that %3D + x) = H (T – x) = -H…

A: Given that H(y)=cosy-siny find H(0), Hπ2, Hπ Show that  Hπ2+x=Hπ-x=-H-x

Q: and as Use the chain rule to find sin(6) r = st, 0 = s + t3 dz as

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Q: 3. Let V = e" cos(qr), with p = x In(xyz), q = z cos(ry) and r = 2 tan-(y). By using the chain rule,…

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Q: 22. Give a formula that gives the length of the outer loop for the curve: r = 4+8 cos(0) (Hint:…

A: To find the length of outer loop of the polar curve

Q: 1. how that { cos(Inx), sin(Inx)} is a fundamental solutions set for x²y" + xy' + y = 0 on the…

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Q: dy dalam dr Gunakanlah diferensiasi parsial untuk menentukan pernyataan untuk kasus-kasus berikut:…

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

Q: 8. Find the length of the given curve over the indicated interval. r(t)= (r', 3 cos(t), 3t) [] 0, 2

A: Given r¯(t)=t2,3cost,3t in the interval [0,π2]

Q: 3. Let V = e cos(qr), with p = x ln(xyz), q = z cos(лy) and r = 2 tan−¹(y). ᎧᏙ By using the chain…

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Q: ind c, and c2 so that y(x) = c, sin x+ c2 cos x will satisfy the given conditions. Determine whether…

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Q: Suppose A(t) = [ e^t 0 0 0 cos t sin t 0 −sin t cos t]…

A: Given: At=et000cos tsin t0-sin tcos t To Find: To show that At-1 exists and then to find it.

Q: Let w = x*y cos s(4z). Find the following.

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Q: Let C, be the rectangle with vertices (0,0). (1,0) (0,2) and (1,2) oriented counterclockwise. Find J…

A: Green's theorem problem....

Q: Ex: If w cos(uv), u = = xyz, y = π 4(x²+y²)' aw find at point (1, 1, 1). əx

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Q: 2. Find the length of the asteroid x = cos't and y=sin't , 0<t<2n. x = cos' t y = sin³ t

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Q: 1/w cos Tº If h(w) = 10 -dz then h'(w) = (m),4

A: Given:-h(w)=∫101wcosx9x2dxTo find: h'(w)

Q: 1) Let 7(t) = (t, 2 cos t, 2 sin t), -4 <t< 4. Find the length of the curve.

A: In

Q: . Solve the following problems: (a) Utt = 0, 0 0, u (x, 0) = x + sin x, Ut (x, 0) = 0, 0 0.

A: Solution

Q: 3. Let u = f(x, y), x = r cos 0 and y = r sin 0. Show that: 1 1 Urr + −Ur + - U00 = Uxx + Uyy. p2 r

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Q: Solve the problem: u – kuxx = e-t sin x , 0 0; satisfying the conditions u(0,t) = 2n, u(t, t) = 4n,…

A: Let the transformation ux,t=vx,t+Ax+B ...............(1)uxx=vxxut=vtthe problem ut-Kuxx=e-t sinx…

Q: 21. ds: C is the quarter-circle r(t) = (2 cos t, 2 sin t), for 0SIST/2.

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Q: Let C be the shortest path that connects t to ni and I = J. (coshz + 1)dz. Then %3D O I=-1-cosh t+…

A: Since you are asking multiple questions. We are answering only the first questions. Let C be the…

Q: 1.Why does the following DE have an unique solution on an open rectangle that contains the point (O;…

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Q: In the case when U = x on the edge y = a (where Yo is a constant), a calculate C, inorder to show…

A: Given at y=a, U=ψ0ax     …Equation(1). And, on substitution of y=a in expression…

Q: Find the length of the curve over the given interval. =8 r = 7(1+ cos 0), 0 <0< 2n

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Q: If z=1+y , I= e' cos(s) and y=e' sin(s), then at (t, s) = (1,0) is as O-2e O 2e O 2e

A: if z=x2+y2 where x=et coss , y=et sins then find ∂z∂s at t,s=1,0

Q: Let r(t) = (sin 3r, cos 3r, /īt), Find T(t) and N(t)

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Q: - Show that u(x,t) e-a't[A sin(2x) + B cos(Ax)] satisfies the PDE uz = a?uxx for arbitrary A, B and…

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Q: . и., — и. — sin x сos t, t>0 В.С.: и(0, г) %3D0, 1.С.: и(х, 0) %3D 0, 1,(x. 0) = 0

A: Solution :-

Q: If z=x+iy=r(cos(0) + isin(e))then

A: The given equation is z=x+iy=rcosθ+isinθ.

Q: Let C be the shortest path that connects t to ni and I = S. (coshz – 1)dz. Then |=-sinh n+ n(1-i)…

A: The solution is given by using line integral method as follows

Q: 23. Give a formula that gives the length of the outer loop for the curve: r= 4-8 cos(0) (Hint: Graph…

A: r=4-8cos(θ)For loops crossedr=04-8cos(θ)=08cos(θ)=4cos(θ)=12θ=π3,5π3(Have symmetry from 0 to π and π…

Q: Suppose r E C[0, 1]. Show that there exists unique y E C[0, 1] such that x(t) = y(t) + 2 5 sin(s +…

A: Consider the provided question,

Q: 1. Eliminate the parameter tgiven x = t – 3 sint and y = 2 + 3 cos t.

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Q: ,3 Let w = x'y cos(2z). Find the following. dw (a) (x, y, z) = (b) (x, y, z) = ду (c) (x, y, z) = dz

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Q: 7. Let F = (2r sin(y)e", x² cos(y)e", a² sin(y)e*). %3D Find along the path c:7 = (sin(t), t,…

A: Given F→=2xsinyez,x2cosyez,x2sinyez. Find ∫CF→·dr→ along the path c:r→=sint,t,cost,-π2≤t≤π2. From…

Q: What values of α and A make y=A cos αt a solution to y′′+3y=0 such that y′(1)=3

A: