Respuesta :
Answer:
m = - 2 is the value of m that the function y= e^mx is a solution of the differential equation, y' + 2y = 0.
Step-by-step explanation:
To determine all values of m so that the function y= e^mx is a solution of the given differential equation.
First, we will find y'.
From, y= e^mx
[tex]y = e^{mx}[/tex]
But, [tex]y' = \frac{d}{dx}y[/tex]
Hence,
[tex]y' = \frac{d}{dx}e^{mx}\\[/tex]
∴ [tex]y' = me^{mx}[/tex]
Now, we will put the values of y' and y into the given differential equation y' + 2y =0
From the question,
[tex]y = e^{mx}[/tex]
and
[tex]y' = me^{mx}[/tex]
Then, [tex]y' + 2y =0\\[/tex] becomes
[tex]me^{mx} + 2(e^{mx}) = 0[/tex]
Then, [tex]me^{mx} = - 2(e^{mx}) \\\\[/tex]
[tex]m = \frac{-2(e^{mx}) }{e^{mx} } \\[/tex]
∴ [tex]m = -2[/tex]
Hence, m = - 2 is the value of m that the function y= e^mx is a solution of the differential equation, y' + 2y = 0.
