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Show that each function $y=f(x)$ is a solution of the accompanying differential equation.$$2 y^{\prime}+3 y=e^{-x}$$a. $y=e^{-x}$b. $y=e^{-x}+e^{-(3 / 2) x}$c. $y=e^{-x}+C e^{-(3 / 2) x}$
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Calculus 1 / AB
Calculus 2 / BC
Chapter 7
Integrals and Transcendental Functions
Section 2
Exponential Change and Separable Differential Equations
Functions
Trig Integrals
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we want to show that these three our solutions or our differential equation Why is he go to why Prime plus three y is equal to e to the negative X was just check to see if they work out or not. So let's first figure out What? Why Prime ms for each of these. So for the 1st 1 there's going to be why Prime is just gonna be negative. E to the negative X. Now we can go out and plug everything in to our differential equation right here. So it's going to be two times why Primes. Let's just multiply this by two and then three times why? And we want to add these two equations together. We're gonna get do you buy? You're right. It same way they have it too. Why Prime plus three by is equal to three minus to eat the negative X which would just be eaten named Rex so that one checks him. Now what about the 2nd 1? Well, this is going to be why Prime is eager to. So it should have the same derivative as First Minister e to the negative X and now this should be negative rehabs e to the negative x And again we would multiply the top one by and we're gonna multiply the bottom one by two. Actually, those two's counts out with each other. Now, when we add these equations so on the left side also have the two by price plus three Y, uh, three e to the negative x plus a negative, too. Heat of the night to extract. Still be e to the negative X and then the e to the negative x over to R E to the negative three House X Well, for those will cancel out with each other, so just be there. So that one check so and then lastly, we can find the derivative. Oh, see? So why pry? There's going to be equal to e toothy. Negative Thanks. Her negative e to the negative x plus. Now it will be negative. Three, huh? See, he too negative. Three x over, too. And just like before, we can go ahead and multiply our first equation by the re smoke three. But me you could read So 33 and three And the next one we're gonna multiply by 22 and two. The twos they're cancel out. And then once again, we get to why Prime plus three y is equal to so three e to the negative X minus two e to the negative extra S B e to the negative X and then the last terms here. Those were just cuts out with each other when we add so that one checks out as well. So it looks like all three of those are solutions of our differential equations.
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