Question

Substitute y(x)= 2 a,x" and the Maclaurin series for 6 sin 3x into y' - 2xy = 6 sin 3x and equate the coefficients of like powers of x on both sides of the equation to n= 0 find the first four nonzero terms in a power series expansion about x = 0 of a general solution to the differential equation.

У(х) = ____ +.
(Type an expression in terms of a, that includes all terms up to order 6.)

Answer 1

Substitute
`y(x)= \sum \limits ^(\infty) _ {n=0}a_nx^n` and the Maclaurin series for 6 sin 3x into y' - 2xy = 6 sin 3x and equate the coefficients of like powers of x on both sides of the equation to n= 0 find the first four nonzero terms in a power series expansion about x = 0 of a general solution to the differential equation.

У(х) = ____ +.

(Type an expression in terms of a, that includes all terms up to order 6.)

Answer:

`\mathbf{y(x) = a_o+(a_o+9)x^2 + ( (a_o)/(2)-(9)/(4))x^4+ ( (a_o)/(6)+ (51)/(40))x^6}`

Explanation:

`y' - 2xy = 6 sin 3x`

`y(x)= \sum \limits ^(\infty) _ {n=0}a_nx^n`

`y'= \sum \limits ^(\infty) _ {n=1}n \ a_nx^(n-1)`

`\sum \limits ^(\infty) _ {n=1}a_nx^(n-1)-2x \sum \limits ^(\infty) _ {n=0}a_nx^(n)= 6(3x- ((3x)^3)/(3!)+ ((3x)^5)/(5!)...)`

Relating the coefficient of constant term

`a_1 =0`

Relating the coefficient of x¹

`2a_2-2a_0 = 18`

`2a_2 = 18+2a_0`

`a_2 = 9+a_0`

Relating the coefficient of x²

`3a_3 - 2a_1 = 0`

`a_3 - a_1 = 0`

`a_3 =a_1`

`a_3 =0`

Relating the coefficient of x³

`4a_4 -2a_2 = -27`

`4a_4 = -27+2a_2`

where;
`2a_2 = 18+2a_0`

`4a_4 = -27+18+2a_0`

`4a_4 = -9+2a_0`

Divide both sides by 4

`(4a_4)/(4) =( -9)/(4)+(2a_0)/(4)`

`a_4 =( -9)/(4)+(a_0)/(2)`

Relating coefficient of x⁴

`a_ 5=0`

Relating coefficient of x⁵

`6a_6 -2a_4 = (6(3)^5)/(5!)`

`6a_6 -2a_4 = (6(3)^5)/(120)`

`6a_6 -2a_4 = (243)/(20)`

`6a_6 = (243)/(20)+2a_4`

Divide both sides by 2

`3a_6 = (243)/(40)+a_4`

where;

`a_4 =( -9)/(4)+(a_0)/(2)`

`3a_6 = (243)/(40)+( -9)/(4)+(a_0)/(2)`

`3a_6 = (243-90)/(40)+(a_0)/(2)`

`3a_6 = (153)/(40)+(a_0)/(2)`

Divide both sides by 3

`a_6 = (51)/(40)+(a_0)/(6)`

Thus:

`\mathbf{y(x) = a_o+(a_o+9)x^2 + ( (a_o)/(2)-(9)/(4))x^4+ ( (a_o)/(6)+ (51)/(40))x^6}`