Download TWK2A Reduction of order (Section 4.2) Problems

TWK2A
Reduction of order (Section 4.2)
Problems
1. If y1 = e5x is a solution of y 00
second solution.
25y = 0; use reduction of order to …nd a
2. If y1 = x4 is a solution of x2 y 00
…nd a second solution.
7xy 0 + 16y = 0; use reduction of order to
3. If y1 = ln x is a solution of xy 00 + y 0 = 0; use reduction of order to …nd a
second solution.
4. If y1 = x2 cos (ln x) is a solution of x2 y 00
order to …nd a second solution.
x
5. If y1 = e 3 is a solution of 6y 00 + y 0
second solution.
3xy 0 + 5y = 0; use reduction of
y = 0; use reduction of order to …nd a
6. If y1 (x) is a solution to a linear homogeneous second-order DE, a second
solution to the equation is given by
R
Z
e P (x)dx
dx;
y2 (x) = y1 (x)
y12 (x)
where P (x) is the coe¢ cient of y 0 in the standard form of the ODE. Show
that y1 and y2 are linearly independent.
7. Consider the equation
ay 00 + by 0 + cy = 0:
b
Given that y1 (x) = e 2a x is a solution to this DE, use reduction of order to
…nd a second solution.