MA2332 Tutorial Sheet: due at tutorial on 11th March, 2016
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1. Consider the second order ODE
x(x − 1)y 00 + 3xy 0 + y = 0.
Use the method of Frobenius to solve for one solution of this equation.
For what values of x is your series solution convergent?
Construct a second solution.
2. Consider the second-order ordinary differential equation
8x2 y 00 + 10xy 0 + (x − 1)y = 0,
(a) Use the Frobenius method to determine the roots of the indicial equation.
(b) Choosing a root of the indicial equation so that the solution is non-singular at
x = 0 determine the corresponding recursion relation.
(c) If the solution is equal to one when x = 1, write down the series solution which
is accurate to within one percent when x < 1.
In each of the following cases find a second solution in the form y(x) = u(x)v(x)
where u(x) is a solution and v(x) is to be determined.
(a) y 00 + 5y 0 + 6y = 0, where one solution is u(x) = e−2x .
Note this is corrected from the version handed out in class where the coefficients
5 and 6 were reversed.
(b) (1 − x2 )y 00 − 2xy 0 = 0, where one solution is u(x) = 1.
Remark: part b) is the α = 0 case of Legendre’s equation.
3. The Legendre polynomials Pn (x) are generated by
∞
X
1
=
hn Pn (x).
Φ(x, h) = √
1 − 2xh + h2
n=0
Write down the first four Legendre polynomials and verify that they are orthogonal,
i.e.
Z 1
Pn (x)Pm (x)dx = 0, n 6= m.
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Sinéad Ryan, see http://www.maths.tcd.ie/˜ryan/231.html
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