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# (a) Show that any function of the form $y = A \sinh mx + B \cosh mx$ satisfies the differential equation $y" = m^2 y.$(b) Find $y = y(x)$ such that $y" = 9y, y(0) = -4,$ and $y'(0) = 6.$

## A. $m^{2} y$B. $2 \sinh 3 x-4 \cosh 3 x$

Derivatives

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he gets close. So when you right here. So we're gonna differentiate using the chain rule so we get the derivative equal to a caution. Mehmet's times. Um, it must be my public function of sign h Amec's times. Mm, which is equal to a M. Karsh of a max was b m the high public function of sign of em. It's we have the second derivative. We're just equal to a M square. My public function of sign of em it's was B m Square cars a mix. So we get this to be equal to and square times A hyperbolic function of sign and necks was B I probably function of co sign annex. So I'm square times a A public function of sign with a max plus B karsh mm is equal to em square a by public function of sign the max. Well, I must be the hyperbolic function of co sign of MX. Refer to be were given Why is equal to a a public function of sign of mx was be cautious of M IX. The second derivative is nine. Why we found the following differential equation. I'm square. Why you make them equal to each other when we get a M value of three were given why of zero is equal to negative for so negative for is equal to a have a heart perfectly function of sline three term zero Must be her public function of co sign three times their own No, we have negative four is equal to a my public phones, you know, sign well zero Let's be coastline of zero and then we get we just plug it in So we get for a sign my probably function of sign zero On the high public function of coastline there's one So we get b is equal to negative form we're gonna use The first derivative of zero is equal to six So we use our first derivative for wine We got six is equal to a three wash three times Earl minus four three sign I probably function three times zero Yeah, A is equal to So when we plug it in and begin why of equal to my prayer of like function of sign Greet X minus four Karsh of three eggs

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