Solve the differential equation below using series...

Solve the differential equation below using series methods. y'' + 2xy' - 5y = 0, y(0) = 4, y'(0) = 2 The first few terms of the series solution are: y = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 Where: a_0 = a_1 = a_2 = a_3 = a_4 =

Transcript

00:01 We are as to solve the differential equation, y double dash plus 2xy dash minus 5y is equal to 0 using the series method.

00:13 And we are given with some initial conditions and they are y of 0 is equal to 4 and y dash of 0 is equal to 2.

00:24 Now let's consider this equation as equation number 1 and these two as equation number 2.

00:34 Understand that the standard equation can be written as yw dash plus p of x times y dash plus q of x is equal to 0.

00:47 If we compare this equation with equation number 1, we can understand that p of x is equal to 2x.

00:55 And q of x is equal to minus 5.

01:01 Clearly, p of x and q of x are analytic at x is equal to 0.

01:09 So what we can say is x is equal to 0 is the ordinary point.

01:16 Now let's assume y to be equal to summation n runs from 0 to infinity, a .n, x to the power n.

01:27 This will be our equation number 3.

01:30 From here, we can compute the value of y -dash.

01:33 It will be equal to summation n runs from 1 to infinity, n times a .n times x to the power n minus 1.

01:45 And similarly, we can compute the value of y double -dash.

01:48 It will be equal to summation n runs from 2 to infinity, n times n plus 1 times a .n times x to the power n minus.

02:00 2.

02:01 From these equation we can get that.

02:07 Summation n runs from 2 to infinity n times n minus 1 times a x to the power n minus 2 plus 2 x times summation n runs from 1 to infinity.

02:23 N times a n times x to the power n minus 1 minus 5 times summation n runs from 0 to infinity .m 5 times summation n runs from 0 to infinity .m...

Question

Solve the differential equation below using series methods. y'' + 2xy' - 5y = 0, y(0) = 4, y'(0) = 2 The first few terms of the series solution are: y = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 Where: a_0 = a_1 = a_2 = a_3 = a_4 =

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