Exercises in ODE
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Transcript of Exercises in ODE
MEGN 502
Advanced Engineering Analysis
Assignment No. 2
1 September 2015
1. Determine the solution for the following ordinary differential equation
d2y
dx2+ x
dy
dx− 4y = 0
using the power series,
y(x) =∞∑k=0
Akxk
Provide a recurrence relation for the Ak.
2. Determine the solution for the following ordinary differential equation(1− 1
2x2)d2y
dx2+ x
dy
dx− y = 0
using the power series,
y(x) =∞∑k=0
Akxk
Provide a recurrence relation for the Ak.
3. Locate and classify the singular points of Problems 1 and 2.
4. Locate and classify the singluar points of the following differential equation
d2y
dx2+ log x
dy
dx+ xy = 0, x ≥ 0
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5. Use the Method of Frobenius to obtain the general solution of the followingdifferential equation valid near x=0:
x(1− x)d2y
dx2− 2
dy
dx+ 2y = 0
6. The differential equation
(1− x2)d2y
dx2− cx
dy
dx+ n(n + c− 1)y = 0
is a specialization of Jacobi’s Equation. Show that if n is nonzero or apositive integer it possesses a polynomial solution of the form
un(x) = 1− n(n + c− 1)
2!x2 +
{n(n− 2)}{(n + c− 1)(n + c + 1)}4!
x4 + · · ·
when n is even, and of the form
vn(x) = x− (n− 1)(n + 2)
3!x3+{(n− 1)(n− 3)}{(n + c)(n + c + 2)}
5!x5−· · ·
when n is odd.
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