By Erwin Kreyszig

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**Extra info for Advanced Engineering Math 9th Edition with Mathematica Computer Manual**

**Example text**

This makes it likely that such a formula should exist. (E) The first two derivatives of y1 and y2 are continuous at x ϭ 0 (the only x at which something could happen). Hence these functions qualify as solutions of a second-order ODE. y1 and y2 are linearly dependent for x м 0 as well as for x Ͻ 0 because in each of these two intervals, one of the functions is identically 0. On Ϫ1 Ͻ x Ͻ 1 they are linearly independent because c1y1 ϩ c2 y2 ϭ 0 gives c1 ϭ 0 when x м 0, and c2 ϭ 0 when x Ͻ 0. The Wronskian is 0 ⅐ 3x 2 Ϫ x 3 ⅐ 0 W ϭ y1y 2Ј Ϫ y2 y 1Ј ϭ { }ϭ0 x 3 ⅐ 0 Ϫ 0 ⅐ 3x 2 xϽ0 if { .

Now solve this by separating variables or as a homogeneous linear ODE. In Prob. 12 we have p ϭ 4; hence W ϭ ce؊4x by integration from 0 to x, where c ϭ y1(0)y 2Ј(0) Ϫ y2(0)y 1Ј(0) ϭ 1 ⅐ Ϫ 0 ⅐ (Ϫ2) ϭ . 7. Nonhomogeneous ODEs. page 78 Purpose. We show that for getting a general solution y of a nonhomogeneous linear ODE we must find a general solution yh of the corresponding homogeneous ODE and then— this is our new task—any particular solution yp of the nonhomogeneous ODE, y ϭ yh ϩ yp. Main Content, Important Concepts General solution, particular solution Continuity of p, q, r suffices for existence and uniqueness.

Euler–Cauchy Equations, page 69 Purpose. Algebraic solution of the Euler–Cauchy equation, which appears in certain applications (see our Example 4) and which we shall need again in Sec. 4 as the simplest equation to which the Frobenius method applies. We have three cases; this is similar to the situation for constant-coefficient equations, to which the Euler–Cauchy equation can be transformed (Team Project 16); however, this fact is of theoretical rather than of practical interest. Comment on Footnote 4 Euler worked in St.