Problem 1. Find the residues of f(z) at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where (z) is given by 1 b) z(1 (1 + z)" (n a positive integer); sin 2z ; е) (z + 1)3

Q: x²y" + xy' + (x² - 1) y = x³/²; y₁ = x-1/2 cos x, y2 = x-1/2 sin x

A: Here we use the method of variation of parameters to find the particular solution of differential…

Q: Evaluate the expression a+b a+b when a=23a=23 and b=45b=45. Write in simplest form

A: # the expressions are printed twice , taking them once given a=23 and b=45 The value of a+b is as…

Q: Question 4. Reduce the equation Uxr + Uyy - 2uz + 4uy + u = 0 into the form Vrr + Vyy +cv = 0 =…

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Q: For what parameters p,q all solutions of y"+py'+qy=0 are periodic functions (y=y(x))

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Q: 3. Use the Fourier-Poisson formula to find the limit as t→ ∞ of the solution to the problem U₁ =…

A: The eigenvalue problem is

Q: y'(0) = 1. y'(0) = -0.5 y'(0) = 0. y'(0) = 7. '0) = 0. )) = 4. )) = 2. y₂ (0)=0. x: (0)1 23) 4y"-8y…

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Q: The homogenous D. E. with constant coefficient of at least order which has the solution y = 7x2 +…

A: Here we have to find ode of given solutions

Q: Ifv= U (x² + y² +2²), then 025 + 0 + 02 is dy² 0 0 01 1 O-1/2 O Other:

A: Here we have to do the partial differentiation of the given function.

Q: 3. Use undetermined coefficients method to find a particular solution of the non-homogeneous ODE y"…

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Q: 2. Consider the problem u"(x) + u'(x) = f(x) u'(0) = u(0) = [u'(1) + u(1)], with f(x) a given…

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Q: By the Existence and Uniqueness Theorem, the IVP y' + y = sin x x2 – 25' y(3) = 0 is guaranteed to…

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Q: Use finite difference approximation to find the values of y(0.25), y(0.5), and y (0.75) for the…

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Q: Let x1(t) = sin(4ät) and x2(t) = 3t. Is x1(t) + x2(t) periodic?

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Q: (a) Define the symbols ôf/DE and af/dz by af 1 (af 10f 2 (dr af 1 (of 18f i by, dz 2 dr' i dy as…

A: Since you have asked multiple questions, we will solve the Question (a) for you. If you want any…

Q: 3) Assume that y, (a) is a solution to y" + P(x)y' +Q(x)y= 0. Prove that -SP(z)dz = - y} (x) y₂(x) =…

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Q: 4) Find ôw/ ôr if w= x=r-s, y= cos(r + s), z = sin(r + s) at the point (r,s)= (1,–1) Ans. 12

A: Find partial derivative using chain rule then put values of x ,y,z in obtained expression then put…

Q: 7) The function sin(7t) 0 8 f(t) = 8t + 9 expressed in terms of the Heaviside function is equivalent…

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Q: Derive the D'Alembert's solutiojn to the following Cauchy problem Un = Uxx, - 0 0 u(x,0) = e,…

A: Consider the Cauchy problem, utt=uxx, -∞<x<∞,  t>0ux,0=e-x, utx,0=sin2x, -∞<x<∞ We…

Q: Show that the stationary path of the functional V S[y] = ["™ dx (y² +y²), _y(0) = 1, y(v) = v, v >…

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Q: 5. If xy + 3y2 = 2, evaluate the values of (a) y at the point (-1, 2) and b Find dy for the…

A: Given: (a) To evaluate y' at the point (-1, 2), (b) To find the derivative dy/dx?

Q: Question 2. Find and classify the singular points of the DE as regular or irregular. (4 Points). а)…

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Q: Q1: Find 1) l (3 + e6t)2 2) e {e-3t(t + 1)²}

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Q: 1. Consider a rectangle with length l and width w, both changing. (a) Show that Δ-w)- ( ) . w+. (Δω)…

A: The definition of ∆lw is∆lw = l+∆lw+∆w - lw                        (1)The graphical representation…

Q: 5. (a) Solve the cauchy problem given by x²ux + xyuy + xu = Xx-y where (x, y) + (0,0) if u(x, xln x)…

A: Given x2ux+xyuy+xu=x-y where x,y≠0,0 if ux,xlnx=-lnx, x>0

Q: 2. Determine whether sin x or cos x is a solution to y" - 2(cot x)y' + (1+2 cot² x) y = 0 Hence use…

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Q: Z= x³+y¹+(x sin y)+ (y cos x),Zx/Zy= (3x²+ sin y-y sin x)/(4y+ xcos y -cos x) O True O False

A: Given Z=x3+y4+(x siny)+(y cosx)To find ZxZy:

Q: 7 ☐ 1. Find the local linear approximation of y = tan x at x = πT 4 О L(x) = 1.117+2.248 x -ㅠ) 7 4 О…

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Q: Let u(g) and v(g) be the Fourier transforms of u(x) and (2) of the product u(z) v(x)? Select one: O…

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Q: 2. (a) Use the method of undetermined coefficient to find the general solution of the non-…

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Q: (b) Following the procedure in (a) to find the eigenvalues A, and A2, write the determinant for the…

A: Since there is (a) is not given completely, hence find the solution of part (b).

Q: Find the equation of the curve whose y" y = sin¯¹(x) — ²√³x - T - 3 6 y = y = sin¯¹(x) - I 2√3 3 y =…

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Q: The singular point(s) of (x² + 4 |(x+ 2)y" + (x- 2)y' - 4y = 0 are A. x = -2 and x = 2 B. All points…

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