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!
Consider the equation xy′′+y′+y= 0, x >0.
a) Verify that 0 is a regular singular point.
(b) Find the indicial equation and its roots.
c) Determine the recurrence relation(you do NOT need to find the
solutions).
(a) Show that x= 0 is a regular singular point.
(b) Find the indicial equation and the indicial roots of it.
(c) Use the Frobenius method to and two series solutions of each
equation
x^2y''+xy'+(x^2-(4/9))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
Determine if x = 0 is an ordinary point, regular singular point, or
irregualr singlar point for the following. Make sure to give
reasons.
a) y" + (2/x)y' + (5e^x)y = 0
b) x(1-x)y" + 4y' + y =0
its
3xy, no y by itself
Determine a differential equation that has x = 0 as a regular
singular point and that the roots of the index equation are i and
i. Find a solution around x = 0.
Given that x =0 is a regular singular point of the given
differential equation, show that the indicial roots of the
singularity differ by an integer. Use the method of Frobenius to
obtain at least one series solution about x = 0.
xy"+(1-x)y'-y=0
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