П Write the function f(x) = x (1-x) as an eigenfunction ex- pansion of the eigenfunctions corresponding to the Sturm-Liouville prob- lem y" + Xy = 0, 0 < x < y(0) = 0 y (H) = 0 πT

Q: Let, f(z) = z-2z +2 (a) Show that, starting from the initial point ro 31 using Newton's method it…

A: Consider the given function.

Q: d/dt (t2 + 1/t + 1)

A: Given ddtt2+1t+1

Q: Ⓒ Solve зик subject to + 2x x = 0 u(x,0) = in Rx [0,00) n(x)

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Q: Perform one iteration of the Newton's method to minimize the function f(x1,x2) = 2(x2 – x?)² + (1 –…

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Q: Find a linear differential operator that annihilates the given function 1+6x-4x3

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Q: 5. Show that --I+#^) log, (V +1- x) + log, (VF+1+x) = 0 x². x² + 1 + x

A: We know the property of logarithm, logb(a)+logb(c)=logb(ac)

Q: Determine whether the functions f1(r) = e2*, f2(x) = e-2" , and f3(x) = sinh 2x are linearly…

A: To determine whether the functions f1x=e2x, f2x=e-2x, and f3x=sinh 2x are linearly independent on…

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3) (2z…

A: 3. Given function is 2z+1zz3-5.

Q: , Find two linearly independent solutions of 2a²y" – xy' + (2x +1)y = 0, x > 0 of the form Y1 = r"…

A: (Note: Please scroll to the last step for the final answer.) Given equation is 2x2y''−xy'+2x+1y=0…

Q: 30. E=2*10^5 I=30000 L=600 W=25 y = dy dx = W 120EIL 3 (-x ² + 21³x² - (^x) W 120EIL (-5X² +…

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Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3)…

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Q: hree solutions of the ODE x³y"" + 6x²y" + 4xy' - 4y = 0 are y₁ = x, y₂ = x-², and 3 = 3x-² + 8x.…

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Q: 23 at the poles (1+i)//2 and (-1+i)//2. Find the residues of 24 +1

A: Let f(z)=z3z4+1 and it has simple poles at 1+i2  and -1+i2. The residue of any f(z) at simple pole…

Q: Orthonormalize [1, t, t²} in P₂(R) with (p, q) = p(0)q(0) + p(1)q(1) +p(2)q(2) }

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Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3)…

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Q: Is the set of functions f₁(x) = ex · f₂(x) = ex- linearly dependent or linearly independent on (-∞,…

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Q: If f is invertible and f( – 1) = 11, then f-'(11) = and (f( – 1))-1 |

A: Given. f(x) is an invertible function. f(-1) = 11

Q: 6) If y=x² +1+ f dt , prove that y"= 2 - x2

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Q: Show that the integral equation TT g(x) = 1 (sin x sin 2t) g(t) dt has no eigenvalue

A: Solution:-

Q: independent solutions of y''+ P(x)y'+ Q(x)y = 0. Assume that f1(x) has at

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Q: E=2*10^5 I=30000 L=600 W=25 y = dy dx W 120EIL = 120 FIL (-5X² + 6(³X²-1²) ū assume as р 38, P = =…

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Q: If e* and xex are solutions of y"tay'tby=- Are soluhons linearly independent? write the soluhion.…

A: Given that e−x and xe−x are the solutions of y''+ay'+by=0. We have to check whether the e−x and xe−x…

Q: (b) Prove that sint -dt > 0 t+1 for all a > 0.

A: Given:- ∫0x sintt+1dt &gt;0

Q: Find two linearly independent solutions of 2x2y"-xy'+(-6x+1)y=0, x>0 of the form

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Q: Let f(x) = (x – b) Ax + c, where A is symmetric and positive definite matrix, b is a vector and c is…

A: Given that  f(X)=X-bTAX+c, where A is symmetric and positive definite matrix, b is a vector and cis…

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (2² + 3z – 5)/z3 (3) (2z…

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Q: 1. Derive the following functions: a) f(x) = 3x - 5x² + 12 b) g(x) 2x³-4x²+2x-1 3-x c) h(x) = cos(x)…

A: Given The functions are given by: a) fx=3x4-5x2+12 b) gx=2x3-4x2+2x-13-x c) hx=cosx x2-3x+2 d)…