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6. Yi Ay2 Y2 = 4y1 I6t2 + 2...

Question

6. Yi Ay2 Y2 = 4y1 I6t2 + 2

6. Yi Ay2 Y2 = 4y1 I6t2 + 2



Answers

$$6 y^{\prime \prime}+y^{\prime}-2 y=0$$

The fact that this expression here we need to multiply our A and R C together. So this will give us 36. Next, we'll find two factors of 36 ton. Multiply together 50 36 but add together to be making it so that is wanting to be negative. Four kinds make it is not. So we're gonna do is we're going to regret this expression but we're going to middle break down the little term into negative for negative. So this will be six y squared minus four. Why minus nine y six now? Because factor by grouping sold on the 1st 2 terms, we're going to factor out too. Why now we'll be left with three. Why? Minus two on the second to terms with, In fact, all the negative three. Now we'll be left with three. Why minus two. So this is equal to three. Why minus two times to why I just

We're simplifying the expression. And to do this, we're going to be doing the distributive property. So I'm gonna take my six, y swear, and I'm gonna multiply it to the first term. So six y squared times. Why gives me six? Why cute. And then against six y squared times too? I swear so. Plus Well, why? To the source.

Here we are given occasion occasion it y two divided web Why police squad left three. Why plus six divided by by it could do for now the simplified if and fall for why we get If I do well, why holy square, The common three That is why bless Good Bad Rabbi is equal to four Now let us as you y plus two divided by y that is a call to X. We get exit squad lefty X Mina four is equal to zero now Fact arrived If we get X plus four x minus one that is equal to zero X plus four is equal to zero R X minus one is equal to zero We get X. It called to mind a for our X equal to one now He called the value off X we get why could divided by wise that is equal to minus four You are why that's good about a rabbi that is equal to one now Solvable dictations. Here we see that practically we thought this this one we get why it's glad playschool is equal to why by square minus y slash two is equal to zero. We get that in the resolution for dedication. So but if Cardiff Now we saw edification. Vice squad, Pashto political to minus four, right, right square Before way. Pashto is equal to zero now. Use that quadratic formula. Kartik for Life X equal to minus B plus minus. Be square Mina four. Easy hold divided by two A. Here a is equal to one B is equal to four and say the call to to use the value of ABC and find the value off why we get why is equal to Pinafore. That's fine of is square Root off B squared that it's 16 pinafore is one and see that it's too who were divided by to. Now, if you get Mina four best minus, so it's quarter to To add it back to final answer it Via is equal to plus minus the square root off to That's the final answer

Okay here were given the equation. Why? Triple crime Mina six wide, double prime by this white prime plus six y is equal to zero. And the characteristic equation is six is R cubed minus six R squared minus R plus six is equal. Does here. And once we factor in this equation this cubic equation we will end up getting are one comma too. I'm a three. It's equal to negative 11 and six So we have three and linearly independent functions here, given by Why Won t sequel to get to the Negative t Why two is equal to each of the tea and why three is equal to eat the 16. And what we can argue here is that we can weaken first conjecture If there are constants, we're not conjecture. We can actually first ask a question if there are constants e b and C that are not all zero. So their constant a bnc. Not all zero such that eight times. Why one. Let's be times wide, too. Let's see Times y three, which is equal to eight times eats the negative t Let's be times e to the T plus C. It was 16 is equal to zero for all tea in the interval given to us which is for all t in negative infinity, some infinity. So we want to try and find out if there are constant A, B and C they're not all zero such. That Disick Wei shin is true for all teeth. But this is not possible. That is equal to B is equal to seize zero since each of the exponential functions are never zero because they're always increasing and the Assam towed towards zero so they don't actually attain the zero function. But they are as methodically tending towards cereals. Tea goes to infinity or not, and depending on the argument off the exponent, which is a function so we can only have this equation be true if a equals b equal secret cereal. And that would mean that f one f two f three armed linearly, independent, our linearly independent on our or negative infinity. Alan. Thank you Disinterest. So we will have that The general solution is given by why Have tea is equal to see one times each The native t let's see two times each the tea let's see three times each of the 16. Whoops. There should be an exponents. So eat to the 16th. So this is our generals. We should


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