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Solve the given problems.For what values of $m$ does the function $y=a e^{m x}$ satisfy the equation $y^{\prime \prime}+4 y=0 ?$

Calculus 1 / AB

Chapter 27

Differentiation of Transcendental Functions

Section 6

Derivative of the Exponential Function

Derivatives

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Lectures

03:09

In mathematics, precalculu…

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In mathematics, a function…

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Determine for which values…

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02:05

For each of the following …

01:25

Determine the function val…

00:33

Determine whether the give…

03:02

Solve each equation for $y…

00:16

Solve each equation.$$…

01:14

Find all values of $a$ suc…

04:07

If $$x M(x, y)+y N(x, y)=0…

00:57

Solve each equation. Write…

Okay. Very similar problem to the last one. This question says four days of em does a that that's why he calls a tense. Either the mx satisfy the following differential equation. Y double prime. This four Y equals zero. So we look at this equation and we split for what derivatives we need. Looks like we need the second derivative. Okay, so let's go ahead and compute that We compete Y prime. Nice quick day. We can compute y double prime. Okay, we're gonna take this into our differential equation and just replace the white level prime with this version of the derivative. Okay, We're gonna do the same thing with the why we can replace the why with what they gave us in the question why he calls a that's either the Mx. Okay, Saudi called a zero. Okay, we can take out the greatest common factor A into the mx should give us m squared plus four equals zero. We can use the zero product property to equate this piece to zero. This will tell this M squared equals negative four. If you're working with real numbers, we're gonna say no real solutions. If we are working with imaginary numbers, we're gonna be able to say m equals plus a minus two. I. Okay. Taking the square root of both sides. We know the square root of negative full is to I and we always have a plus or minus when we put a square reading. So depending on what you're working with real numbers, real solutions, if you are using imaginary and complex numbers, you can have em equals plus a minus two. I.

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