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!
Use an appropriate infinite series method about
x = 0
to find two solutions of the given differential equation. (Enter
the first four nonzero terms for each linearly independent
solution, if there are fewer than four nonzero terms then enter all
terms. Some beginning terms have been provided for you.)
y'' − xy' − 3y = 0
y1
=
1
+
3
2
x2 + + ⋯
y2
=
x
+
+ ⋯
Find the infinite series for the following differential equation
about x = 0, using Frobenius method, Bessel's or Legrende's
equations.
x^2y" + 4xy' + (2+x)y = 0
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