Answer :
Answer:
The set of solutions are linearly independent because the wronskian is
e^(2ax) ≠ 0
Step-by-step 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.