Case Study Research: Design and Methods, 3rd Edition by Robert K. Yin

By Robert K. Yin

This new version of the best-selling Case examine Research has been conscientiously revised, up-to-date, and improved whereas preserving nearly all the positive aspects and insurance of the Second Edition. Robert Yin's complete presentation covers all points of the case examine method--from challenge definition, layout, and information assortment, to information research and composition and reporting. Yin additionally strains the makes use of and value of case stories to quite a lot of disciplines, from sociology, psychology and historical past to administration, making plans, social paintings, and education.

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21) we have x1 u = and v = log x. 23) . 24) , such that their Wronskian is unity. We are concerned with finding one— parameter groups of the form (3. 23) invariant. 25) . 25) From this condition we deduce to be zero. 25) becomes d3y __! &_. - + g 5 3 g 4 g Using this equation and 3 dy1 dy1 i— + dx1 + q(x1)y1 = 0 I we obtain on multiplying by — g2 the equation, + + — 4gtg" + p(f)gg' + q(f)g3]y = o. 23) is self—adjolnt (see problem 14) is a constant. 29). general solution of zero. 30) , and we have — where A, B = 4(AC — B2) + and denote arbitrary constants and we have used the fact that C the Wronskian of for a given q(x) = where E(x) I is unity.

12) . 13) . 14) , are the canonical coordinatee of the group and i8 eome function of u only. 6) we need to find = (x+x2—y2)fl — Unfortunately = Try 1 and (that is, (x+x2+y2) n +y and n(x,y) such that ,,2 — (x+x2+y2)2 — + must now be determined by trial and n(x,y) constant) then F error. must satisfy — (x+x2—y2) = Unfortunately even at this stage we cannot systematically solve this equation since the solution by Lagrange's method involves solving the original differential equation (see problem 3).

This result is due originally to Lie [1] (see problem 1) and is generally given some prominence in the literature. However, from the point of view of actually solving differential equations the use of canonical coordinates is preferable. Moreover as we have seen in the previous chapter canonical coordinates can be used with higher order equations and therefore we will emphasize their use here. 9) we see that there exists a function 9z_ — —F 9z ' z(x,y) 1 9y — (ri—FE) . 11) 51 But we have az dx dz = dy—Fdx dy =0 = Thus where we have used (4.

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