Let $$ y^{\prime \prime}-9 y=0 $$. Find all values of $$ r $$ such that $$ y=k e^{r x} $$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list. $$ r= $$ Submit answer Next item (numbers)

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Given the differential equation

$$
y^{\prime\prime} - 9y = 0,
$$

we assume a solution of the form

$$
y = ke^{rx}.
$$

Step 1: Compute the first and second derivatives of $$ y $$:

$$
y' = k r e^{rx},
$$
$$
y'' = k r^2 e^{rx}.
$$

Step 2: Substitute $$ y $$ and $$ y'' $$ into the differential equation:

$$
k r^2 e^{rx} - 9ke^{rx} = 0.
$$

Step 3: Factor out the common terms $$ ke^{rx} $$ (noting $$ ke^{rx} \neq 0 $$):

$$
ke^{rx}(r^2 - 9) = 0.
$$

Since $$ ke^{rx} \neq 0 $$, we must have:

$$
r^2 - 9 = 0.
$$

Step 4: Solve the equation $$ r^2 - 9 = 0 $$:

$$
r^2 = 9.
$$

Taking the square root of both sides, we get:

$$
r = 3 \quad \text{or} \quad r = -3.
$$

Thus, the values of $$ r $$ are:

$$
r = 3, -3.
$$
$$ r = 3, -3 $$

Let .
Find all values of such that satisfies the differential equation. If there is more than
one correct answer, enter your answers as a comma separated list.

Submit answer Next item (numbers)

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Let . Find all values of such that satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list. Submit answer Next item (numbers)