Homework 6 (MATH 2310-04) Name (Print):
Due date: Tuesday, March 24, 2009
1. Determine whether the given functions form a fundamental set of solutions.
a) f() = cos(2) 2 cos2(); g() = cos(2) + 2 sin2().
b) f(t) = et cos(t); g(t) = et sin(t); 0.
:Solution
a) No.
b) Yes.
2. Consider the differential equation
.0)1('y,1)1(y,0y9'y8''y
a) Find two solutions for this equation.
b) Calculate the Wronskian to show that these two solutions form a fundamental set of
solutions.
c) Calculate the solution for the initial value problem by adopting the Wronskian for
the calculation of c1 and c2.
:Solution
a) y1(t) = e9(t1), y2(t) = e t1
b) W(y1, y2) 0
c) y(t) = 0.1 e9(t1) + 0.9 e t1.
3. Consider the differential equation
.0)1('y,1)1(y,0y)2x('y)2x(x''yx2
a) Verify that two solutions are given by y1 = x, y2 = x ex.
b) Calculate the Wronskian to show that these two solutions form a fundamental set of
solutions.
c) Calculate the solution for the initial value problem by adopting the Wronskian for
the calculation of c1 and c2.
:Solution
a) y1 = x, y2 = x ex
b) W(y1, y2) 0
c) y(x) = 2 x x ex / e.