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Problem: initial value Solve the 9(0) =-- (12 pts) #4: 2et Problem V -y = 60. solution a5 I of the behavior . the Determine...

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Problem: initial value Solve the 9(0) =-- (12 pts) #4: 2et Problem V -y = 60. solution a5 I of the behavior . the Determine

problem: initial value Solve the 9(0) =-- (12 pts) #4: 2et Problem V -y = 60. solution a5 I of the behavior . the Determine



Answers

In Exercises $47-60,$ solve the initial value problem. $$ \frac{d y}{d t}=t+2 e^{t-9}, y(9)=4 $$

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About the second on the Audi A on the farm. I wiped dry Times Square, double prime plus B times quite cramp. Let's see why Echo +20 From here, we can determine defied the characteristic equation. I asked where, plus B plus C e Co +20 Now suppose in this case we consider the Katrina repeated route I want equal to the art. You then assertion why we go to the constancy one. Eat you the, uh, condition we could You are so uh, X then plus C two times x times e to the x in this question were given, uh, a why nine Wind up a prom, plus try and white prom Plus for why equal to zero with the initial value wiser equal to one. And while prime zero ico 20 here the first time we need to write down the different characteristics Question. Now I ask where plus trail a plus for equal to zero. From here we noticed that this one will be the perfect square after three r plus two square equal to zero. This means then, uh, it was equal to the minus two at the three. So what The repeated route here we have the sole solution. Why it will you go to the constancy one into the minister at the three x plus the constancy to you a time x times each two by minus two out of three ex And now it would try to use the initial value here. Why the, uh it we go to Nah see one. And this one We go to zero here, so we should get to see one and equal to one. And now, when we compute the white prime, when you go to the minus two out of three C one age of the monastery of three acts and then plus C two each of the managed two out of three ex Onda plus a man ist two out of three c two x each of the by Minister of the three X And from here it will compare the white prime of the zero. You get Nico too the manager two out of three C one and then this plus two And this one equal to zero here on this is equal to zero. Therefore, it means that the C two equal to the two out of three C one and equal to the two out of three. Therefore, the final answer will be why it we go to see one ICO too. Ah one. So we can it will equal to the each environment is two of the three x c two equal to the two out of three x times. Each environment is two of the three x.

Okay to solve this initial value problem. First, we're going to put this. Ah, or we're going to find the complementary solution first. So that's when we set the right hand side of this equation equals zero. So why I see men equal to zero. This has Theo. Auxiliary equation P of art. Um is equal to R minus one times ar minus two times are minus three. Is he going to zero? So this has the roots. 12 and three. So are complementary. Solution is gonna have the form. Why of C is equal to C one e to the X plus C two e two the two X plus C three e to the three X. Now we need Teoh. Take care of this right hand side here. So that means that we have f of X is equal to six e to the four X. So this four here means that we have an r is equal to four. So our Annihilator a of D is going to be D minus four now. This reroute here is not included in our roots here. So our trial solution is just gonna have the form. Why? P of X is going to be equal to a not he to the four X like so. So our general solution Why of X? It's going to be equal to C one e to the X plus C two e two the two X plus C three needs the three X, um three x plus A not eat to the forex Now to solve for a not we're going to plug in art Y p into this here. So if we plug in, why P into our original equation here? So we're gonna be taking D minus one a man D minus to D minus three of a not eat of the four x that symbol through six feet of the four x. So applying this first operator to the function here you have d minus one D minus two and then the derivative of this is gonna be four a not e to the four X and then we also have a minus or X minus three a. Not e to the forex. Secret is six feet of the four X. This can then simplify into just for a ministry. So we just have a not e to the four X Okay, so we'll replace that with just a not eat of the four X like so Now applying this operator to this function here. Now you get so for a not either four x again. But this time we have minus two a not e to the forex for X. Okay, so then we have Ah, let's go to six feet of the four X. So we have four a not minus two A. Not so then Now. Ah, this just becomes to a not eat of the four X. So now, one more time we apply this so we're going to get eight a not e to the four X and then minus two a not e to the four X. We get six feet of the four X get six. A not eat of the four X is equal to six feet of the four X so the heat of the four X can cancel out, and then we just have six a equal six. So a not is gonna be able to one. So a nonsensical the one. So this is going to be our ah, general solution now. So now we're going to use our initial values to solve for C one, C two and C three. So then we need to find why prime of X first. So this is going to be able to c one e to the eggs plus two C two e two the two x plus three C three e to the three X and then plus E four e to the four X. Okay, let me plus four e to the forex. Let me move all of this for E to the forex. Now we need to also find why double prime of X thing is gonna be able to see one e to the X plus four c two e two the two X plus nine C three e to the three X plus 16 each of the four X. Now we need to, uh, use our or plug in our three initial mission. So why of zero? This is equal to C one plus C two plus C three plus one is equal to four. Then we're gonna have why, prime of zero, this is gonna be C one plus two c two plus three C three plus four. It's gonna be equal to 10 and then finally. Why? Double prime of zero. It's going to be equal to C one plus four C two plus nine C three plus 16 is equal to 30. So we can now. Ah, let's see here from the first and second equation. Ah, we can move this. Well from all of equations, we can first move this over to the other side. So we have This is gonna be equal to three. Uh, let me write it like this. Um, Then we move this over to the next page. So you have C one plus C two plus C three. It's gonna be able to three. Then we also have C one plus two C two plus three C three. It's gonna be equal to six. Then C one plus four C two plus nine C three. It's and be able to 14 uh, 14 se. Um, yeah. 14. There. Okay, so then now we're going to, um Well, we can subtract this for equation from this here. If we do that, then we get C two plus two. C three is equal to three. Like so. Okay. Ah, actually, all right, we're going to just solve this using um, augmented matrix. Actually, we'll just put it into Okay. 123 149 And then we have 36 14 so we can get rid of the first and second column here by subtracting the first column. So we're gonna get 111 three, and then three 01 to than zero. Then we're gonna have four minus one is three. So we'll have three here, and then nine minus one is going to be eight. Will have eight here. And then, um, 14 minus three is 11. So we have 11 here, then this can go further. So we're going to use this one to get rid of this one and this three here. Okay, so subtracting here, we get 10 negative one negative one and then three minus three is enemy zero. Then we keep ah, this the same 01 to you. And then now we're going to subtract three here, So eight minus, uh, eight. Minus six is going to become, uh, to lumps to And then now, um, 11 minus, ah, nine. That's gonna be also to hear. So the first or the last row here tells us that C three is equal to three. Thanks. So we can further reduce set into one and one. Like so, uh, before that, actually, we can subtract off this here. Okay? Eso then becomes zero one. Then we have 11 Then we can add this road to this row here. So then it becomes zero 111 So we get that C two c c one C two C three. All are equal to zero. So we're final solution. Why of X? This is going to be C one or sorry soon be just either X plus he to the two x plus heat to the three X need to the three X and then plus, um, each of the four X So this is going to be our final solution here.

Yeah. Hello. The question is taken from first total linear differential equation and equation is divided over the X is equal to x plus y. And the initial condition is why a tax is equal to zero is equal to minus of left hand side can bear it on S. T. O. By over the x minus Y is equal to x. Okay how we can solve it we can solve it by evaluating the integrated factor. Integrated factor is equal to exponential Integration. The coefficient of this way that is -1 into DFX. That is able to exponential minus affect that. Just multiply the whole equation with this integration integrated sector we get exponential minus X into divide over the x minus exponential minus X into why that is able to X exponential minus of x. Okay so left hand side can we get an S. D. Over the X. Why exponential minus of X? That is equal to x. Exponential minus of X. Taking d X to the right hand side. We get the Y to the power minus X is equal to act into the power minus X into the X. And integrating so left hand side is like integration of one with respect to the quantity inside the bracket. That is called Y exponential minus X. Physical too integration of uh taking access for us function and exponential minus access second function. So we that is equal to x. Exponential minus X into minus or one. So they're very good minus x. X. Exponential minus x minus integration of differentiation of facts that this one and integration of exponential minus affects. That is exponential minus six and minus one plus sees the constant of integration mm So this becomes by exponential minus X is equal to minus x. Exponential minus X. And the integration of exponential minus X is minus exponential minus X. Let's see. Using the initial condition. Initial condition is why a tax is equal to zero is minus four. Okay let me substitute the value of x zero. In this equation we get minus four is equal to zero minus one plus seats From here C is equal to minus of three. Substituting the value for seeing the equation we get by exponential minus X is equal to exponential minus x minus of and inside the bracket X plus one plus ses minus of dividing the whole equation with explanation minus x X. We get Y is equal to minus x plus one -3. Exponential. Just that requires solution of the equation. All this clears your doubt and thank you


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