Assignment 1
Due date: July 14, 2015
1. Solve y 0 = 2xy 2 and y(1) = 1.
2
2. Find the solution to y 0 − 2xy = ex and y(0) = 0.
3. Let α be a real constant, find a fundamental set of solutions to y 00 − 4y 0 + α(4 − α)y = 0. (Hint: You need to
consider two cases which depend on α.)
4. Find the general solution to y 00 + 10y 0 + 25y = 0.
5. Find a fundamental set of solutions to 2y 00 + 4y 0 + 9y = 0.
6. Find a particular solution of y 00 − y = 2 sin(x2 ).
7. Find the solution to x2 y 00 + 6xy 0 + 6y = 0, y(1) = 0 and y 0 (1) = 1.
8. Find the general solution to x2 y 00 + 5xy 0 + 4y = 0 for x > 0.
9. Find the general solution to x2 y 00 + 3xy 0 + 2y = 0 for x > 0.
10. Let α, β be constants, find the condition that α and β should satisfy for which all solutions of x2 y 00 + (α +
1)xy 0 + βy = 0 for x > 0 approach zero as x → 0. (Hint: lim xr ln x = 0 for all r > 0.)
x→0
11. Find the power series solution about x = 1 to the problem y 00 + 4y 0 + 6xy = 0, y(1) = 0 and y 0 (1) = 1.
12. Consider the differential equation 2(x − 1)y 00 + y 0 + y = 0 for x > 0.
(a) Find the general power series solutions at x = 0 (find the first four non-zero terms). What should be the
minimal radius of convergence of this series?
(b) Find the power series solutions at x = 0 (find the first four non-zero terms) such that y(0) = 0 and
y 00 (0) = 2.
13. The Chebyshev differential equation is
(1 − x2 )y 00 − xy 0 + α2 y = 0,
where α is a real constant.
(a) Determined two linearly independent solutions in powers of x for |x| < 1.
(b) Show that if α is a non-negative integer n, then there is a polynomial solution of degree n.
(c) Find a polynomial solution for each of the case α = n = 0, 1, 2, 3.