Question

find the value of m such that the function y=e^mx
is asolution of y`+2y=0

Answer 1

Do you know what y' is?

Answer 2

If so, perform that operation on y=e^(mx).

Answer 3

y` is the first dirvative for y with respect to x i perform that operation on y but i cant continue

e^mx*(m+2)=0

Answer 4

If you have

\(a*b=0\)

then either \(a=0\) or \(b=0\)

Answer 5

\(\color{#000000}{ \displaystyle y=e^{mx} }\)
\(\color{#000000}{ \displaystyle y'=me^{mx} }\) (via chain rule)

\(\color{#000000}{ \displaystyle y'+ny=0\quad \longrightarrow \quad me^{mx} +n(e^{mx} )=0\quad \longrightarrow \quad (m+n)e^{mx} =0 }\)

Using the Zero Property (if I recall the terminology correctly),
\(\color{#000000}{ \displaystyle 0=e^{mx} }\) or \(\color{#000000}{ \displaystyle 0=m+n }\)

Then you know that:
\(\color{#000000}{ \displaystyle e^{mx}\ne 0 ~~\forall\{m,x\} }\), and \(\color{#000000}{ \displaystyle m+n=0\quad \Longrightarrow \quad \color{red}{m=-n} }\)