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Numerade Educator

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Problem 1 Easy Difficulty

Show that $ y = \frac{2}{3}e^x + e^{-2x} $ is a solution of the differential equation $ y^x + 2y = 2e^x $.

Answer

$$=\frac{6}{3} e^{x}=2 e^{x}=\mathrm{RHS}$$

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Video Transcript

What's the show? That this is a solution to our deferentially, please. So, first, we're going to take the derivative of dysfunction here. So why Prime equals two birds ender got a movie to the exits. Just needs to be it now for a second. Luke. Clues derivative. Ah, eats the negative. Two x is a derivative of this here, which is negative too. All times e to the negative two weeks so we can rewrite this as 2/3 need to be X minus two times e ds and negative to it. Now we need to somehow show that this equation and it's equal to this equation. But first, we need to get rid of this y variable here by plugging in our original Why equation that was given to us. So we're going to say 1st 1 to subtract two are from both sides. So have negative too. Why plus two times each Be X now we're gonna play gin are y equation. So why pride equals negative, too. Times 2/3 e to the x plus e to the negative to x all plus two times e to the X. Now British retards who will have negative 4/3 usually X minus two times E to the negative two ex all place two times in two New Yorks. Now, this is almost similar to this here, which is what we want. However, we can combine like terms because these two have the same base here. So too, if we give it a denominator of three, our top will become six because an original denominator is one what's part of top and bottom by three. So we get 6/3 I'm gonna be right there. Says 6/3 here and now we can combine thes two times here. So we have 6/3 minus 4/3. That gives us positive 2/3 into the X and then bring down our two times eats of negative to X. And because this here is equal to this here, we know that it is a solution.