Question

I need the solution of the question 3.

Q1. Find a power series solution in powers of x.

y''+(1+x^2)y=0

Q2. Find a solution of (1-x^2)y''-2xy'+n(n+1)y=0 by reduction to the Legendre equation.

Q3. Find a basis of solutions. Try to identify the series as expansions of known functions.

(1) xy''+2y'-xy=0

(2) (x^2+x)y''+(4x+2)y'+2y=0

Answer #1

Find the power series solution around x=0. Find the first few
nonzero terms of each solution. Power series are not necessary to
solve try older methods in addition to power series.
1) y”+x^2y=0

Find at least one solution about the singular point x = 0 using
the power series method. Determine the second solution using the
method of reduction of order.
xy′′ + (1−x)y′ − y = 0

1) Solve by power series around x=0: y"-2xy'-2y=0
(Find the first three nonzero terms of each of the LI
solutions)

Find the first 5 non-zero terms of power series
(1-x^2)y''-2xy'+2y =0

Find at least four non-zero terms in a power series expansion of
the solution to the initial value problem:
y'' + xy' + e^x y = 1-x^3
y(0) = 1, y'(0) = 0

find the minimum convergence radius of the solutions on power
series of the differential equation (x^2 -2x+10)y''+xy'-4y=0
surrounding the ordinary point x=1

slove ODE using power series and find the radius of
convergence
(1-x^2)y''-2xy'+2y=e^(2x)

Let Q1, Q2, Q3, Q4 be constants so that y =Q1+Q2x+Q3x^2+Q4x^3
satisfies that y(1)=1 and (1-x^2)y"-2xy'+12y=0.

Find two solutions of a power series for the differential
equation y'' - xy = 0 surrounding the ordinary point x=0

Find the general solution near x = 0 of y'' - xy' + 2y = 0.
(Power series, recursive formula problem)

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