What is the largest interval (if any) on which the Wronskian of y1 e10-2t and y2 = 1 is non-zero? e2t O (0, 1) O (-1,1) O (0, ∞0) O (-00, 00) 1 is equal to zero The Wronskian of y1 = el0-2t and y2 e2t everywhere.

Q: 2. (a) Calculate (i+1)302 + (i − 1)302 and write it in the form of x + iy. (b) Find an upper and…

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Q: The domain of the function f(x) = cos-¹(e) is O(-00, 1] O(-∞0,0] O(-∞o, In(cos 1)] ○ [0,00) O[-1,1]

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Q: (6) Let f(a), g(n) € Pla] where f(a) is irreducible.

A: Let m = deg(f(x)) and n = deg(h(x)).

Q: Show limn→∞(1 +1/2+1/3+···+1/n−lnn) exists

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Q: %3D

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Q: Consider the first order ODE y' = √y² – 9. Use the fundamental existence-uniqueness theorem (on page…

A: The given first order ODE is .The aim check whether the ODE has a unique solution or not at the…

Q: The Hermite polynomials Hn(x) are special functions that are extremely im- portant in quantum…

A: For first proof differentiate exponential function and for second part Differentiating series

Q: d Prove: If z = zo is a zero of order n of f(z), then it is zero of order 2n of 2 (f(z))²

A: If a complex number z0 is a zero of order n of the function f(z), it means:f(z0) = 0All derivatives…

Q: A cubic spline S is defined by: So(x) = 3(x- 1) + 2(x – 1)² – (x – 1) S(x) = s,(x) = 4 + 4(x – 2) –…

A: Natural Cubic Spline is a piece-wise cubic polynomial that is twice continuously differentiable.…

Q: 2. Find the largest open subset ECR2 for which (a) f(x) = (b) f(x) = x₁ is continuously…

A: Given Data: We have to find the largest open subset.

Q: Prove that if p(x) is a monotonic increasing positive func tion and M[p(X)] = m exists, then P(X >…

A: The above inequality is an application of Markov-Chebyshev inequality.

Q: 5 Is the function f(x) = (-1)" X[n,n+1) n21 integrable on R in the Lebesgue sense. Here xA is the…

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Q: 1. Let f, g, and h be defined on [0, 1]: f(x) = g(x) = h(x) = 0 whenever x is irrational; f(x) = 1…

A: Given: f,g, h be defined on [0,1] as  fx=1   x∈ℚ0   x∈ℚc  gx=x     x∈ℚ0     x∈ℚchx=1/n       x=mn∈ℚ…

Q: 5) Compute ln(z) for the following complex numbers under the given branch cut. (a) In ( 1/1 + ²√³)…

A: Given that:a)  with a branch cut given by the curve .b)  with a branch cut given by the curve ,To…

Q: Use Fourier-Motzkin elimination to give a linear combination of the following in- equalities that…

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Q: A second order Caushy – Euler nonhomogenous having a solution y = c\cos(4Inx) + c2sin(4Inx) + +Inx…

A: The given solution is y=c1cos(4lnx)+c2sin(4 ln x)+14lnx check the equation x2y''+xy'+16y=4lnx if we…

Q: x = 10 (s+2) (s³+ s)'

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Q: Let f(x) be a function f:[−1,1]→[0,1], f(x)=x^2. Let g(x):(−∞,∞)→[0,1] be a function such that g(x)…

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Q: 25.3 Let X = (Xn)n≥o be a submartingale with Doob decomposition Xn Xo+Mn + An. Show that E{An} <∞,…

A: Let X=Xnn≥0 be a sub martingale with Doob decomposition Xn= X0+Mn+An. To show that EAn&lt;∞ each…

Q: + ao be monic, separable, 2. Let f(x) = E[x], f(x) = x² + An-1xn-1 which splits over E. Let ☀ €…

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Q: Use Ro ucne s Theorem to determine now zeros OI the given iunctions Ile 1n the given annulus: (a)…

A: a Let fz=z4-2z-2 and gz=2 Now on the circle z=12 fz+gz=z4-2z-2+2…

Q: Compute the suitable integral representation of the following function 0, 0.5, —2 2

A: The given, G1x=0,x&lt;-20.5,-2&lt;x&lt;-11+x,-1&lt;x&lt;01-x,0&lt;x&lt;10.5,1&lt;x&lt;20,x&gt;2

Q: 7) The step function (r) is defined as 0, if x a. Show that 0(2) admits the integral representation…

A: As per the question we are given the step function θ(x-a) and we have to prove the given integral…

Q: 1 m # n için | Pm(x) Pn(x)dx = 0 %3D -1 b-) show that note: Legendre equation (1 – a²)Pm (x) – 2æPm…

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Q: Example 19-29. If U, is a rational function of the second degree, prove that : b+c \a-b)(a – c) (c +…

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Q: verify the set {∅n = cos(nx) (∞ n=0) is orthogonal on the interval [−pi, pi]

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Q: Use the Residue theorem to evaluate the integral fe(z-2)dz.

A: Use the expansion of e1z  =1 + 1z + 12! z2 + 13! z3 + ...  Hence ∫e1z(z-2)dz = ∫(z-2)1 + 1z + 12! z2…

Q: Q1. Expand the following function along 1 < |Z| < 3 1 f(z)= (Z+1)(Z+3) Hint: Use partial fraction

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Q: Let n be a natural number and let y1, Y2, Y3, . . . , Yn be real numbers. Prove by induction that…

A: (a) Consider n∈ℕ, and y1,y2,…,yn are real numbers. Let Pn:y1-yn≤y1-y2+y2-y3+…+yn-1-yn. Consider P2,…

Q: The function f is defined by 1 1 - exp(z)* f(z) Show that f has poles of order 1 at the points z =…

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Q: Find all second-order three-step methods in the form Yn+1 = aoyn+q₂yn-l + a₂yn-2 +h[b_f (tn+1• Yn+1)…

A: Second-order three-step methods in the form you provided are used to solve initial value problems…