Answer Happy

Find a power series expansion about x=0 for a general solution to the given differential equation. Your answer should include a general formula for the coefficients. −z′′+x2z′+xz=0 A. z(x)=a0​(1+n=1∑∞​(−1)n(3n)!(1⋅4⋅7⋯(3n−2))2​x3n)+a1​(x+n=1∑∞​(−1)n(3n+1)!(2⋅5⋅8⋯(3n−1))2​x3n+1) B. z(x)=a0​(1+n=1∑∞​(3n−2)!(3⋅6⋅9⋯(3n))​x3n)+a1​(x+n=1∑∞​(3n+2)!(1⋅4⋅7⋯(3n−2))​x3n+1) c. z(x)=a0​(1+n=1∑∞​(−1)n(3n−2)!(3⋅6⋅9⋯(3n))​x3n)+a1​(x+n=1∑∞​(−1)n(3n+2)!(1⋅4⋅7⋯(3n−2))​x3n+1) D. z(x)=a0​(1+n=1∑∞​(3n)!(1⋅4⋅7⋯(3n−2))2​x3n)+a1​(x+n=1∑∞​(3n+1)!(2⋅5⋅8⋯(3n−1))2​x3n+1)
Find the first four nonzero terms in a power series expansion about x=0 for a general solution to the given differential equation. w′′−12x2w′+w=0 w(x)=+⋯ (Type an expression in terms of a0​ and a1​ that includes all terms up to order 3 .)
Find the first four nonzero terms in a power series expansion about x=0 for the solution to the given initial value problem. w′′+6xw′−w=0;w(0)=2,w′(0)=0 w(x)=+… (Type an expression that includes all terms up to order 6.)