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1) Since P(x) = or Q(x) = are not analytic at x = 0, x = is a singular point of the differential equation. Using F heorem, we must check that xP(x) = and x? Q(x) = are both analytic at x = 0. Since xP(x) and x² Q(x) are analyt = 0 is a regular singular point for the differential equation 7xy" - y + 10y = 0. From the result of Frobenius' Theorem, we may assume xy" – y + 10y = 0 has a solution of the form y = x' E Cnx" which converges for x E (0, R) where r and R are constants that will be c n=0 ater.

1) Since P(x) = or Q(x) = are not analytic at x = 0, x = is a singular point of the differential equation. Using F heorem, we must check that xP(x) = and x? Q(x) = are both analytic at x = 0. Since xP(x) and x² Q(x) are analyt = 0 is a regular singular point for the differential equation 7xy" - y + 10y = 0. From the result of Frobenius' Theorem, we may assume xy" – y + 10y = 0 has a solution of the form y = x' E Cnx" which converges for x E (0, R) where r and R are constants that will be c n=0 ater.

Advanced Engineering Mathematics
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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7xy" – y + 10y = 0. (1) Since P(x) = or Q(x) = are not analytic at x = 0, x = is a singular point of the differential equation. Using Frobenius' Theorem, we must check that xP(x) = and x? Q(x) = are both analytic at x = 0. Since xP(x) and x² Q(x) are analytic at x = 0, x = 0 is a regular singular point for the differential equation 7xy" – y + 10y = 0. From the result of Frobenius' Theorem, we may assume that 00 7xy" – y + 10y = 0 has a solution of the form y = x" E C,x" which converges for x E (0, R) where r and R are constants that will be determined n=0 later. (2) Substituting y = x" , Cn x" into 7xy" – y + 10y = 0, we get that 00 + n=0 The subscripts on the c's should be increasing and numbers or in terms of n. (3) In this step, we will use the equation above to find the indicial roots and the recurrence relation of the differential equation. (a) From the equation above, we know that the indicial roots of the differential equation are (in increasing order) r = and r = (b) From the series above, we find that the recurrence relation is for
Transcribed Image Text:7xy" – y + 10y = 0. (1) Since P(x) = or Q(x) = are not analytic at x = 0, x = is a singular point of the differential equation. Using Frobenius' Theorem, we must check that xP(x) = and x? Q(x) = are both analytic at x = 0. Since xP(x) and x² Q(x) are analytic at x = 0, x = 0 is a regular singular point for the differential equation 7xy" – y + 10y = 0. From the result of Frobenius' Theorem, we may assume that 00 7xy" – y + 10y = 0 has a solution of the form y = x" E C,x" which converges for x E (0, R) where r and R are constants that will be determined n=0 later. (2) Substituting y = x" , Cn x" into 7xy" – y + 10y = 0, we get that 00 + n=0 The subscripts on the c's should be increasing and numbers or in terms of n. (3) In this step, we will use the equation above to find the indicial roots and the recurrence relation of the differential equation. (a) From the equation above, we know that the indicial roots of the differential equation are (in increasing order) r = and r = (b) From the series above, we find that the recurrence relation is for
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