If e-x,xe-x are solution of y" + ay + by = 0, then sum of the values of a and b is _________.Correct answer is '3'. Can you explain this answer?

Question

Answer

To find the solution of y, we need to substitute e^(-x) and xe^(-x) into the differential equation y'' + y = 0.

Let's start with e^(-x):

y = e^(-x)
y' = -e^(-x)
y'' = e^(-x)

Substituting these into the differential equation:

(e^(-x))'' + e^(-x) = 0
e^(-x) + e^(-x) = 0
2e^(-x) = 0

This equation does not hold true for all values of x, so e^(-x) is not a solution of y'' + y = 0.

Now let's try xe^(-x):

y = xe^(-x)
y' = e^(-x) - xe^(-x)
y'' = -e^(-x) + xe^(-x)

Substituting these into the differential equation:

(-e^(-x) + xe^(-x)) + xe^(-x) = 0
-xe^(-x) + 2xe^(-x) = 0
xe^(-x) = 0

This equation holds true for x = 0, so xe^(-x) is a solution of y'' + y = 0.

Therefore, the solution of y is xe^(-x).

Answer

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