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 x Q(x) are analytic at x = 0, x = 0 is a regular singular point for the differential equation 7xy" – v + 10y = 0. From the result of Frobenius' Theorem, we may assume that 7xy" – y + 10y = 0 has a solution of the form y = x" > Cnx" which converges for x E (0, R) where 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 x-l+ = 0 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 %3D (4) The general solution to 7xy" – y + 10y = 0 is y = Ax x2+ 3 + ... x+ +В х x²+ x + ... + x+ and converges on the interval
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 x Q(x) are analytic at x = 0, x = 0 is a regular singular point for the differential equation 7xy" – v + 10y = 0. From the result of Frobenius' Theorem, we may assume that 7xy" – y + 10y = 0 has a solution of the form y = x" > Cnx" which converges for x E (0, R) where 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 x-l+ = 0 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 %3D (4) The general solution to 7xy" – y + 10y = 0 is y = Ax x2+ 3 + ... x+ +В х x²+ x + ... + x+ and converges on the interval
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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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 x Q(x) are analytic at x = 0,
x = 0 is a regular singular point for the differential equation 7xy" – v + 10y = 0. From the result of Frobenius' Theorem, we may assume that
7xy" – y + 10y = 0 has a solution of the form y = x" > Cnx" which converges for x E (0, R) where
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
x-l+
= 0
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
%3D
(4) The general solution to 7xy" – y + 10y = 0 is
y = Ax
x2+
3 + ...
x+
+В х
x²+
x + ...
+
x+
and converges on the interval
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