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

Q: 2. Consider the PDE u; - t ux+tu=0 for - o <x< ∞ and 0 <t< ∞ with u (0, x) = sine x. Determine u (t,…

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Q: 4. Find a solution of the Bessel's equation of order zero.

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Q: 2. Solve the Initial - Boundary Value Problem, Ии — а?ии — 0, for 0<x< 1, 0 <t <o∞ u(0, t) и(1,t) %—…

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Q: 1. Show that c(t) = (e¬t cos t, e-t sin t), 0 <t <∞ has finite length and calculate its valu

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Q: In Exercises 27-30, find the differential. 27. d(Vī-x) 28. d(e* + x) 29. d(arctan 4x) 30. d(8" + x)

A: 27. d1-x2 Ans.  ddx1-x2 =ddx(1-x2)12                         =  121-x212-1× ddx(1-x2)…

Q: • Exercise 4. Solve the initial value problem Y' = AY, Y(0) = Y, by computing etAY, where (2 1 1…

A: Given A=211110101,Y0=11-1. solve the IVP.

Q: In a neutron cell opening and closing of ion channels is governed by following equation: du a (1 —…

A: Consider the given differential equation.

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Q: 92 that the elect ric potential prove -5x V = e Cos 13y sinh 12 Z achieves the Laplace equation اثبت…

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Q: b) Show that the “energy function" E(x, y) = 5 + is non-increasing along the solution, i.e., using…

A: (b) The objective is to show that the energy function  is non increasing along the solution, that…

Q: Let z = cosh t. Find the differential dz when t = In 3 and dt = 0.2 O 4/3 O 4/15 O 8/15 O 1/3

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Q: 18. Make a rough sketch of a direction field for the autonomous differential equation y' = f(y),…

A: Consider the sketch of the autonomous differential equation,.

Q: -2+z) ds over t e point (0, 0, 0)

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Q: 10. Find the linearization, L(x,y,z) of the function f (x, y, z) = e* + cos(y + z) at Po (0,",").

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Q: 5. The motion of a mass attached to a dashpot and a spring as it experiences a force can be…

A: Given differential equation is  md2xdt2-Bdxdt-kx=1    , m=1, B=1, k=6, x0=0, x'0=0⇒d2xdt2-dxdt-6x=1…

Q: Assume that after months, butterflies’ populace in 1000 is calculated as: f(w) = 12w + cos(16w)…

A: Given f(w) = 12w + cos(16w) +n * 100

Q: 2(3) Calculate the following commutators: (a) [d/dx, x·d/dx] (b) [sin(x), cos(x)] (c) [x^2, Ĥ] ,…

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Q: Verify Young's Theorem y? x2 z = In(2x + y) – + e(x+2y) + - y

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Q: A homogeneous string is stretched and its ends are at x = 0 and x 1. Motion is started by displacing…

A: The boundary conditions are y(0,t) = y(l,t) = 0 and initial conditions are y(x,0) = μ0sinπxl and…

Q: 23. Given that f(0) = v2 cos (e-), the minimum value of o1+ v2 C. %3D COS D.-1 A. V2-1. B.2N2

A: Since you have posted multiple question in this question so I have solved first question as per our…

Q: 4. Solve the Cauchy problem Utt = 9uxx+cos x, (x, t) E Rx (0, ∞) u(x,0)=0, TER u(a,0)=0, x ER using…

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Q: 1. Consider the function z = x In(1+ y²), if x 2-t and y= Vf, use an appropriate form %3D dz at t…

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Q: 1 Find the local linear approximation of f(x) = , at 3+ 1 Tg = 2, where Az = I- 2. NOTE: Enter the…

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Q: az az and as Use the chain rule to find at (=) Z = tan V u = 8s + 5t, v = 5s – 8t az as az at

A: Topic:- derivatives

Q: 3- h(x, y, z) = cos xy + e*y + In zx, Po (1,0,1/2), A = i + 2j + 2k

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Q: Study the following delay differential equa- tion (Find conditions on c1 and c2 to in- vestigate the…

A: If C1≥C2, Then the steady state : N*=1C1+C2 is asymptotically stable, for all delay τ≥0

Q: az and as az Use the chain rule to find at z = In(4x + 3y), x = s sin(t), y = t cos(s) az as az at

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Q: Suppose that w = x² + y² – z² | And x = psin(4)cos(0), y = psin(4)sin(0), z = pcos(4) Use…

A: Simplify w by substituting values of x , y and z and then solve the partial derivative.

Q: A) If w = 2 y et* – In z where x = In(t² + 1), | dw y = tan-1 3t and z= e3t. Find by Chain Rule dt

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

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Q: 8. Let f(x, y) = cos(y – x) and r(t) = (e', e2t). Find if (r(t)) at t = ln(3).

A: The solution is given as

Q: SSS zdV Q3: Evaluate E + y² + z² = 1 and x² + y² + z² = 4 in the upper where E lies between the…

A: We have to evaluate the integral ∭EzdV, where E lies between the spheres x2+y2+z2=1 and x2+y2+z2=4…

Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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Q: The differential operator (D2 - 16D+113)³ annihilates the functions

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Q: |1) Use a sine expansion to compute the solution of ( ди д'и at ôx? u(0,t)=u(1,t) = 0,t > 0, u(x,0)…

A: According to the given information, it is required to compute the solution of the given equations.

Q: Consider the following simultaneous equations model. Assume that x is exogenous. yi = Bx + e Y2 =…

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Q: 3. Use chain rule to find given w = Vx2 +2y2 + z² and a cos (st), y = -2s In t ane 2 = 3s?Vt. 3s2Vt.

A: Follow the procedure given below.

Q: Let M, = Z, – Z;-1, Zo = 0, where {Zn, n21} is a martingale. Show that E(M,„Z„-1] = 0

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Q: 4] I Let w = 5 cos(xy) – sin(xz); x = 1/t, y = t, z = dw using an appropriate form of the chain…

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Q: Find the Wronskian for y, = sin 3x , and y2 = cos 3x.

A: The Wronskian, W, of two functions, f and g is calculated as follows. W=fgf'g' , where, f' , g'are…

Q: Example 6 Solve the inhomogeneous ODE dy(t) + 3y(t) = e', dt with the initial condition, at t = 0, y…

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Q: Verify Young's Theorem z = In(2x + y) y? x2 +e(x+2y) y

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Q: Consider d?x = -kx ax3 Assume k > 0, m> 0, ¤ > 0, and o > 0. Show that as t → ∞, x → 0. [Hint: Form…

A: According to the given information, consider the given differential equation:

Q: If (х, у) %3D х" у"is an integratingfactor of the the DE. (1+ 3ху)dx + (ху-1 +x?)dy %3 0 Then, the…

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Q: 1- Find dwi dt at the given value for w ==+2,x = cos (1), y = sin (1),z = w =-+ at = 3

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Q: Q1) Suppose w =÷+ ?, x = cos² t, y = sin² t, z = ; Evaluate dw at t = 3, by using the Chain Rule dt

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