Question

Find the value of m so that the functiony=emx is a solution of the differential equation y′+2y=0

Answer 1

Answer on Question #69667 – Math – Differential Equations

Question

Find the value of $m$ so that the function $y = e^{mx}$ is a solution of the differential equation $y' + 2y = 0$.

Solution

Since the function $y = e^{mx}$ is a solution of the differential equation $y' + 2y = 0$, then $\left(e^{mx}\right)' + 2e^{mx} = 0$.

Thus,

$$\left(e^{mx}\right)' + 2e^{mx} = 0,$$ $$me^{mx} + 2e^{mx} = 0,$$ $$e^{mx}(m + 2) = 0.$$

Since $e^{mx} > 0$ for any real numbers $m$ and $x$, then $m + 2 = 0$.

Therefore, $m = -2$.

Answer: $m = -2$.

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