Question

Find all values of m so that the function

y = emx
is a solution of the given differential equation. (Enter your answers as a comma-separated list.)

y' + 7y = 0

Answer 1

Answer:

$m = -7$

Step-by-step explanation:

The objective is to find all values $m$ so that the function $y=e^{mx}$ is a solution of the differential equation $y'+7y =0$.

If $y=e^{mx}$ is a solution of the given differential equation, then it and its first derivative must satisfy the given equation. Let's calculate the derivative.

                      $y = e^{mx} \implies y' = e^{mx} \overset{\text{Chain Rule}}{\cdot} (mx)' = me^{mx}$

Substituting $e^{mx}$  for $y$ and $me^{mx}$ for $y'$ in the equation gives

                          $me^{mx} + 7e^{mx} = 0 \iff e^{mx}(m+7) = 0$

We can divide both sides by $e^{mx}$, since $e^{mx} > 0, \; \forall x,m \in \mathbb{R} \implies e^{mx} \neq 0$. Thus,

                                     $m +7 = 0 \implies m = -7$

Therefore, the function $y = e^{-7x}$ is a solution of the differential equation $y'+7y = 0.$