Question

Show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent.

{eax, xeax}, a ≠ b
Find the Wronskian for the set of solutions.
{eax, xeax}, a ≠ b

Answer 1

The set of solutions are linearly independent because the wronskian is

e^(2ax) ≠ 0

Explanation:

To show that the set of solutions {e^(ax), xe^(ax)} are linearly independent, we find the Wronskian of the set of solutions.

Wronskian of y1 and y2 is given as:

W(y1, y2) = y1y2' - y1'y2

y1 = e^(ax)

y1' = ae^(ax)

y2 = xe^(ax)

y2' = axe^(ax) + e^(ax)

W(y1, y2) = e^(ax)[axe^(ax) + e^(ax)] - ae^(ax).xe^(ax)

= (ax + 1)e^(2ax) - axe^(2ax)

= e^(2ax)

The wronskian of the given set of solutions is e^(2ax), which is not zero.

Because it is not zero, the set of solutions are linearly independent.