find the values of k for y= e^kx satisfies the equation y+y'=y"

Answer 1

Substitute in y= e^kx:

y' = ke^kx

y" = k^2e^kx

Now we can write the equation as:

e^kx + ke^kx = k^2e^kx

Combine like terms:

(1+k-k^2) e^kx = 0

For this to be true for all x, we must have:

1+k-k^2 = 0

This is a quadratic equation in k, so we can solve using the quadratic formula:

k = (-1 ± √5)/2

Therefore, the values of k that satisfy the equation y= e^kx and y+y'=y" are:

k = (-1 + √5)/2 and k = (-1 - √5)/2.

Answer 2

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Answer 3

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