Use power series to find two linearly independent solutions
centered at the point x=0
1) y'' + 2y' - 2y = 0
2) 2x2y'' + x(x-1)y' - 2y = 0
please show work, thank you!
Two linearly independent solutions of the following
equation
(1 − x) y″ +
x y′ − y = 0
are y1(x) =
4ex and
y2(x) = 8x.
(a)
Find the Wronskian
W(y1, y2) of
y1 and y2.
(b)
Using the method of variation of parameters, find a particular
solution of
(1 − x) y″ +
x y′ − y =
2(x −
1)2 e −x
Use the method of Frobenius to obtain two linearly independent
series solutions about x = 0. Form the general solution on
(0,inf).
2x^2y'' - xy' + (x^2 + 1)y = 0
a) Find the Taylor series for sinh(x) (centered at x=0), for e^x
(centered at x=0) and hyperbolic sine and hyperbolic cosine.
b) same as a but cosh(x) instead