Use the Frob e nius m e th o d to solve: x y \" - y = 0 . F ind i nd e x r a nd

Question

Use the Frobenius method to solve: xy"- y = 0 . Find index r and recurrence relation. Compute the first 5 terms (a 0 - a4 ) using the recurrence relation for each solution and index r.

Explanation / Answer

The method only produces one solution. Sometimes the method gives two solutions, but one is all that the theorem regarding the method of Frobenius guarantees. Put y = ? a_n x^(n+r) with the sum from n = 0 to infinity. Then y'' = ? ? a_n (n + r)(n + r - 1) x^(n + r - 2) n=0 Multiplying y'' by x, and factoring out the common x^r gives xy'' - y = . . ? x^r ? [(n + r)(n + r - 1) a_n x^(n - 1) - a_n x^n] = 0 . . n=0 We can divide out the x^r term to make it more manageable. Then, take the n = 0 term out of the first part, and reindex that part (the part corresponding to xy'') by letting n --> n + 1. ? ?{ [(n + 1 + r)(n + r) a_(n+1) - a_n] x? } + a_0r(r - 1)x^(-1) = 0. n=0 The indicial equation (I wrote in on the right side for convenience) is r(r - 1) = 0 ==> r = 0 or r = 1. As it turns out, both produce the same solution. The recurrence relation for r = 0 is (n + 1)n a_(n+1) = a_n for n ? 0. When n = 0, the left side is zero giving a_0 = 0. The coefficient a_1 is a free variable. For n ? 1 a_(n+1) = a_n/(n(n + 1)). a_2 = a_1/2 a_3 = a_2/(2

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.