Substitute [tex]y(x)= \sum \limits ^{\infty} _ {n=0}a_nx^n[/tex] 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:
[tex]\mathbf{y(x) = a_o+(a_o+9)x^2 + ( \dfrac{a_o}{2}-\dfrac{9}{4})x^4+ ( \dfrac{a_o}{6}+ \dfrac{51}{40})x^6}[/tex]
Step-by-step explanation:
[tex]y' - 2xy = 6 sin 3x[/tex]
[tex]y(x)= \sum \limits ^{\infty} _ {n=0}a_nx^n[/tex]
[tex]y'= \sum \limits ^{\infty} _ {n=1}n \ a_nx^{n-1}[/tex]
[tex]\sum \limits ^{\infty} _ {n=1}a_nx^{n-1}-2x \sum \limits ^{\infty} _ {n=0}a_nx^{n}= 6(3x- \dfrac{(3x)^3}{3!}+ \dfrac{(3x)^5}{5!}...)[/tex]
Relating the coefficient of constant term
[tex]a_1 =0[/tex]
Relating the coefficient of x¹
[tex]2a_2-2a_0 = 18[/tex]
[tex]2a_2 = 18+2a_0[/tex]
[tex]a_2 = 9+a_0[/tex]
Relating the coefficient of x²
[tex]3a_3 - 2a_1 = 0[/tex]
[tex]a_3 - a_1 = 0[/tex]
[tex]a_3 =a_1[/tex]
[tex]a_3 =0[/tex]
Relating the coefficient of x³
[tex]4a_4 -2a_2 = -27[/tex]
[tex]4a_4 = -27+2a_2[/tex]
where; [tex]2a_2 = 18+2a_0[/tex]
[tex]4a_4 = -27+18+2a_0[/tex]
[tex]4a_4 = -9+2a_0[/tex]
Divide both sides by 4
[tex]\dfrac{4a_4}{4} =\dfrac{ -9}{4}+\dfrac{2a_0}{4}[/tex]
[tex]a_4 =\dfrac{ -9}{4}+\dfrac{a_0}{2}[/tex]
Relating coefficient of x⁴
[tex]a_ 5=0[/tex]
Relating coefficient of x⁵
[tex]6a_6 -2a_4 = \dfrac{6(3)^5}{5!}[/tex]
[tex]6a_6 -2a_4 = \dfrac{6(3)^5}{120}[/tex]
[tex]6a_6 -2a_4 = \dfrac{243}{20}[/tex]
[tex]6a_6 = \dfrac{243}{20}+2a_4[/tex]
Divide both sides by 2
[tex]3a_6 = \dfrac{243}{40}+a_4[/tex]
where;
[tex]a_4 =\dfrac{ -9}{4}+\dfrac{a_0}{2}[/tex]
[tex]3a_6 = \dfrac{243}{40}+\dfrac{ -9}{4}+\dfrac{a_0}{2}[/tex]
[tex]3a_6 = \dfrac{243-90}{40}+\dfrac{a_0}{2}[/tex]
[tex]3a_6 = \dfrac{153}{40}+\dfrac{a_0}{2}[/tex]
Divide both sides by 3
[tex]a_6 = \dfrac{51}{40}+\dfrac{a_0}{6}[/tex]
Thus:
[tex]\mathbf{y(x) = a_o+(a_o+9)x^2 + ( \dfrac{a_o}{2}-\dfrac{9}{4})x^4+ ( \dfrac{a_o}{6}+ \dfrac{51}{40})x^6}[/tex]
