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. $$ r= $$

Question

UpStudy AI · Accepted Answer

We are given the differential equation:

$$
y'' - 9y = 0
$$

Assume a solution of the form:

$$
y = ke^{rx}
$$

Then the first derivative is:

$$
y' = kre^{rx}
$$

and the second derivative is:

$$
y'' = kr^2 e^{rx}
$$

Substitute these into the differential equation:

$$
kr^2 e^{rx} - 9(ke^{rx}) = 0
$$

Factor out $$ke^{rx}$$ (which is never zero):

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

This implies:

$$
r^2 - 9 = 0
$$

Solve for $$r$$:

$$
r^2 = 9 \quad \Longrightarrow \quad r = 3 \quad \text{or} \quad r = -3
$$

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.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions

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.