Question

Find all values of m so that the function y= e^mx is a solution of the given differential equation.

Answer 1

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

$y = e^{mx}$

But, $y' = \frac{d}{dx}y$

Hence,

$y' = \frac{d}{dx}e^{mx}$

∴ $y' = me^{mx}$

Now, we will put the values of y' and y into the given differential equation y' + 2y =0

From the question,

$y = e^{mx}$

and

$y' = me^{mx}$

Then, $y' + 2y =0$ becomes

$me^{mx} + 2(e^{mx}) = 0$

Then, $me^{mx} = - 2(e^{mx})$

$m = \frac{-2(e^{mx}) }{e^{mx} }$

∴ $m = -2$

Hence, m = - 2 is the value of m that the function y= e^mx is a solution of the differential equation, y' + 2y = 0.