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

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.

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step-by-step explanation:

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0.6, 3/5, and 6% are the 3 same portion.

but, when they're all changed to a decimal they would equal to 0.6.

hope this , good luck!