1. Show for Dirac spinors u(p) and v(p) that E u9 (p)a (p) = (p + m) ab and v (p)o (p) = (p – m) .b i=1,1 i=f,!

Q: 3. Find the solution of the following linear system by Operator method or Laplace transform method.…

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Q: Let x(t) be a Brownian motion with x(0). Mathematically prove or disprove that -4x(t)is a Brownian…

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Q: 1. - Solve the below equation using Laplace transform: y" -y=t, y (0) = 1. y' (o)=1 fin 96) 1/s 21/²…

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Q: Which of the following is the Laplace transform of the solution to the initial value problem x" –…

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Q: The linearization at a = 0 to v8 + 5x is A+ Bx. Compute A and B. A = B = ||

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Q: { Vo, æ L,

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Q: Define: T:P, → R by T(P) = [ p(x)dx. What is the kernel of T ?

A: Given that: T:P2→RTp=∫01px dx

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f + 3g, v= fg, find the Wronskian of u…

A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

Q: If la,} and {b„} are divergent then {a, + b,} is divergent. True False

A: Given: an and bn are divergent . The objective is to state whether an+bn is divergent is true or…

Q: Give the Laplace transform of the solution to [p' +2y =4 e** – 5 sin(5 x), y(0) =-3] 25 3 а) Y(s) =…

A: Option (a) is true.

Q: 29. In Theorem 9.8, if f'(x o) = 0, no conclusion can be drawn about the equilibrium point xo of x'…

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Q: Let (B; ) be a Brownian motion with Bo = 0. Define a process (Z,) by Z, = B{l/t for t > 0. What is…

A: Given information: It is given that Bt is a Brownian motion with B0 = 0.

Q: In Exercises 23–30, find Tn centered at x = a for all n. 26. f (x) = e", a =-2

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Q: Give the Laplace transform of the solution to [y " + 2y'+5y=4x+e*, y(0) =4, y '(0) =3] 4 1 4 (5+ 2)…

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Q: Consider the ARMA(2,1) process (1+0.6L)(1+0.2L)y, = (1+0.5L)e, . (a) Construct an autocorrelation…

A: To determine:  ACF of ARMA(2,1) process Given: (1+0.6L)(1+0.2L) yt= (1+0.5L)et

Q: 1 prove that | e -ym dy = mr(m) -ут

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Q: 1) Find the norm of the following operators d) T:C[0,1] → C[0,1] by (Tx)(t) = f¿(t - s)*x(s)ds

A: Norm of operator gives the idea of size of any linear operator. Norm of can be found as .

Q: * Let A E R"Xn, ĩ E R" \ {0} and A = 0. Prove that if the origin is stable if and only if ī is…

A: Given A∈ℝn×n,x¯∈ℝn\0 and Ax¯=0 To prove that the origin is stable if and only if x¯ is stable.  If…

Q: Consider the following semilinear first-order PDE ди ди +y ду = xe (i) Find the general solution to…

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Q: Sow that 2 x² + y² + z² - c²t² is Lorentz invariant.

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Q: Looking Ahead Use the Wronskian to show that {e' , te' I is linearly independent on R.

A: Consider the equation

Q: -) Find the eigenvalues and the normalized.cigenfunctions of the following boundary value problem:…

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

Q: Show that the functions e*, cos x, and sin x are linearly independent using the Wronskian.

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Q: 8. Prove that the differential operator M: C² [a,b] → C[a,b] defined by d²u du Mu= 2+3u dx is a…

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Q: 3.4.10. (a) Given x E co, prove that there exist vectors xn E co such that ||x – xn|| → 0 as n → o.

A: As you have asked more than one different questions, we shall answer first question only..   For…

Q: Consider h(x,y, z) = cos (xy) + eY² + In (xz) at the point P ( 1,0, = 1 Deter-

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Q: 4. Evaluate the Kullback-Leibler divergence between two Gaussians p(x) = N(xu, E) and %3D q(x) =…

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Q: 1- is The differential of u= In 1+ t2 (A du = tª – 1 lot 2t at В du = none of the given choices D 2…

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Q: у" + 4y' — 12у %3D —15е-3X У(0) %3 3 dan У'(0) — 9

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Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: 4. Let z = x2 + y³, x = 2st, y = s - t2, s = u", and t = u sin w. Compute for Oz ду and дw ди

A: Let z=x2+y3, x=2st, y=s-t2, s=uw, and t=u sin w.To compute for ∂z∂w and ∂y∂u.

Q: What is the Laplace transformed version of this ODE a" + 3x' + 9x = e=t with initial conditions a…

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Q: Please solve this differential equation question. thanks!

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Q: Give the Laplace transform of the solution to [y" +3 y' +5 y=5 e 3*, y(0) = 4, y'(0) =2] 5 14 a) O…

A: We will find out the required value.

Q: 2. Evaluate ř(t) dt: 4 4t ř(t) t2 + 5t + 6' î + 2e4t cos t j - k; a = 0 and b = 4 t2 + 1

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Q: Suppose the goind densiy Runction Xanef y i ginen by frxy) = 44 (ス-8) ○ZXC0,○ ス %3D otheraise…

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Q: Q6. Let C[0,1] have the inner product < f,g 2 S,"f(x)g(x)dx and let fn= cos nx (n=0,1,2,3,....).…

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Q: 19: Find two linear operator T, and T,on V(R) such that T,T, = 0 but T,7, - 0.

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Q: (2) Given that "dw and ewz +e twz = 2 cos(wz), we obtain by applying the inverse Fourier Transform:…

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Q: Show that e", sin x, cos a are linearly independent using Wronskian Method.

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Q: -52 +1 -3 If zo =-1, G(z) = and f (y) = -5 y1 4 - 2 y2 Y3 + v -4 y1 - 2 32, find the Jacobian of G…

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Q: Suppose w = In(æ² + y² + z²) , Compute dw/du and dw/dv in two different ways: ue3" sin(9v) , y :…

A: GIvenw=ln(x2+y2+z2) where x=ue3vsin(9v) , y=ue3vcos9v and z=ue3vTo find∂w∂u and ∂w∂v

Q: 1) Let s,T: e defined by Sx = (0,x1, X2, - ) Tx - (x2,X3, Xyy ..) .. Show that (i) Ilsil = TII = 1.…

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Q: 13. Use the chain rule, in Leibniz notation, to find dx dy at the given value of x. а. у %3D Зи? —…

A: We are solving the first three subparts and for the next subparts post the question againand also…

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

A: ANSWER:

Q: 1 20 В -2 1 x = с 3 –10 X, 1 3 a) Calculate the pseudoinverse B* . X1 for Bx = c, call it x. X2 b)…

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Q: Give the Laplace transform of the solution to [y " + 2y'+ 5y=0, y(0) =4, y '(0) =3] 4 (5+ 2) 3 + a)…

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Q: Solve the De as follows via Laplace transformations: 18y" + 27y" + 8y' + 12y = 0; y(0) = 19/12,…

A: The solution is given as

Q: 1. On (–1,1), two functions Y1 (x) = e2« cos x, y2(x) = e2ª sin x are linearly independent linearly…

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

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.