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Cobbles · Answer 1 · 2021-11-14T22:11:06+0000

Answer:

m = -7

Explanation:

The objective is to find all values
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
and
me^(mx) for
in the equation gives

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

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

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

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

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