4) Use Stirling's approximation to find an approximate formula for the multiplicity of atwo-state paramagnet. (You can assume that N₂ << N).

Answered: Image /qna-images/answer/5800b476-0b5a-4be8-a0fd-1756a03ba6a6.jpg

Answered: 4) Use Stirling's approximation to find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5. Solve the DE dy d²y xdx² dx - 4x = 0, 1
20
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2.2) The basis of the equation y" + 2y" = 0 ,are: a) 1,1, e2x Q.3) The radius of convergence of (x-1)y" + 4y' + 6xy = 0 about xo = 2, is: a)+o Q. %3D b) 1,1,e-2x c) 1, x, e2x d) 1,x, e-2x %3D b) 2 c) 1 d) 0
Denote three scalars A = eTE-le, B = µ™E-le and C = µTE-'µ. 1. Prove that AC – B² > 0.
The differential equation of the family of curve y = ax +b+1, where a, b are arbitrary а, constant is d²y dx d²y b) dx +y = 0 d) none of these dx a) +y = 0 c) O d) a) O b) O c)
Use the table to the right to answer the following question Energy es hem Energy released by buming 1 kg of a certain combustble sedimentary rock Energy released by fesion of 1 kg of a certain ge Energy released by fusion of hydrogen in 1 ter of water How many kilograms of combustible sedimentanry rock would be required to supply the energy needs of country A for 1 year? Country A annual energy consumption Country Bannual erergy consumption Rwould take approximately kilograms. (Use scientific noton. Use the multiplication symbol in the math palete as needed. Round to three decimal places as needed)
8) At a particular point on a given space curve, r'=v= (8,4,-8), r'=a3D(0,6,12), and N=(1,2,2). apa Write a as a linear combination of T and N. (Note that this is an "orthogonal decomposition" of a.) bb). ) What is the tangential acceleration a, at this point? c)t What is the radius of curvature at this point (this is the reciprocal of the curvature)?
2
18) Determine the annihilator of y=z-esinz. a) (D-2D³-2D²)y=0 b) (D + 2D³ + 2D²)y=0 e) (D¹+2D³-2D²)v-0 d) (D¹-2D³ +2D²)y=0
Q2) Classify the following system of first-order partial differential equations (y=const.): aw aw = u ax at ди ди +y at ax Which second-order pde would this first-order system sorrespond to ? (Hint: Damped wave eguation or special case of the Telegrapher equation).
a) Show that the fourth-order Runge-Kutta method, k₁ = hf (ti, wi), h k₁ k₂= hfti+ Wi + 2 2 h k3 = hfti + Wi + 2¹ 2 k4 = hf (ti + h, wi + k3), 1 W₁+1 =W₁ + (k₁ + 2k2 + 2k3+k4), when applied to the differential equation y' = Ay, can be written in the form 1 Wi+1 = = (1 + hλ + 1 (hx)² + = (ha)³ + 2/7 (hX)^) wi- Wi. 24

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4) Use Stirling's approximation to find an approximate formula for the multiplicity of a two-state paramagnet. (You can assume that N₁<< N).