Question

**Series Solutions Near a regular singular point:**
Find two linearly independent solutions to the given differential
equation.

3x^{2}y"-2xy'-(2+x^{2})y=0

Answer #1

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7. Given that x =0 is a regular singular point of the given
differential equation, show that the indicial roots of the
singularity do not differ by an integer. Use the method of
Frobenius to obtain two linearly independent series solutions about
x = 0. Form the general solution on (0, ∞)
2xy”- y’ + y = 0

Given that x =0 is a regular singular point of the given
differential equation, show that the indicial roots of the
singularity do not differ by an integer. Use the method of
Frobenius to obtain to linearly independent series solutions about
x = 0. Form the general solution on (0, ∞)
3xy”+(2 – x)y’ – y = 0

Differential Equation:
Determine two linearly independent power series solutions
centered at x=0.
y” - x^2 y’ - 2xy = 0

Solve the given differential equation by means of a power series
about the given point x0. Find the recurrence relation; also find
the first four terms in each of two linearly independent solutions
(unless the series terminates sooner). If possible, find the
general term in each solution.
y′′ + xy = 0, x0 = 0

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'' − 2xy' − y = 0

Series Solutions of Ordinary Differential Equations For the
following problems solve the given differential equation by means
of a power series about the given point x0. Find the recurrence
relation; also find the first four terms in each of two linearly
independed sollutions (unless the series terminates sooner). If
possible, find the general term in each solution.
y"+k2x2y=0, x0=0,
k-constant

Series Solution Method. Solve the given differential equation by
means of a power series about the given point x0. Find the
recurrence relation; also find the first four terms in each of two
linearly independent solutions (unless the series terminates
sooner). If possible, find the general term in each solution.
(1 − x)y′′ + y = 0, x0 = 0

The point x = 0 is a regular singular point of the differential
equation. x^2y'' + (9 /5 x + x^2) y' − 1/ 5 y = 0. Use the general
form of the indicial equation (14) in Section 6.3 r(r − 1) + a0 r +
b0 = 0 (14) to find the indicial roots of the singularity. (List
the indicial roots below as a comma-separated list.) r =

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
+

The indicated functions are known linearly independent solutions
of the associated homogeneous differential equation on (0, ∞). Find
the general solution of the given nonhomogeneous equation.
x2y'' + xy' + y = sec(ln(x))
y1 = cos(ln(x)), y2 = sin(ln(x))

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