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EXERCISES 12.3In Exercises 1-14 find the general solution of the differential equation. dy 1. &x +3y=0 2 dy dx (sinh x)y = 0 3 dy +2y = 4 dy dx dx 2y = * dy 5. ...

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EXERCISES 12.3In Exercises 1-14 find the general solution of the differential equation. dy 1. &x +3y=0 2 dy dx (sinh x)y = 0 3 dy +2y = 4 dy dx dx 2y = * dy 5. &x-ay = f(x) 6. y + 2xy = 4x where f is continuous 1. y + 6xsy=xs Y33 (xy + 1) for x > 0 9.Y-y=1ze 10. d + ycosX = COsX 1L. dy + ytanx = tan x 12 dy ytan x = esin* dx dx for -w/2 < x < t/2 for -w/2 < x < t/2 13. dr {Y ~ sint? = 0 14. dy+y=tsint? + St dt for t > 0 for t > 0 In Exercises 15-18 fnd the particular

EXERCISES 12.3 In Exercises 1-14 find the general solution of the differential equation. dy 1. &x +3y=0 2 dy dx (sinh x)y = 0 3 dy +2y = 4 dy dx dx 2y = * dy 5. &x-ay = f(x) 6. y + 2xy = 4x where f is continuous 1. y + 6xsy=xs Y33 (xy + 1) for x > 0 9.Y-y=1ze 10. d + ycosX = COsX 1L. dy + ytanx = tan x 12 dy ytan x = esin* dx dx for -w/2 < x < t/2 for -w/2 < x < t/2 13. dr {Y ~ sint? = 0 14. dy+y=tsint? + St dt for t > 0 for t > 0 In Exercises 15-18 fnd the particular solution satisfying initial condition:



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Differential Equation In Exercises 123 and 124 , find the particular solution that satisfies the initial conditions.
$$
\begin{array}{l}{f^{\prime \prime}(x)=\sin x+e^{2 x}} \\ {f(0)=\frac{1}{4}, f^{\prime}(0)=\frac{1}{2}}\end{array}
$$

Calculated differential equation by simply doing the integrations prime access. We, um, integrated from some Some point can actually get the X. All right, So, um, this would gives us decision to the minus minus two minus x plus C one. And we know to have prime zero is zero trees, one minus c. So this is zero. You see, steel, how we gonna see one is to right? So? So then we can do that integration once again. It's just Dana Gordon of a Pro, Max, which is we want to. So remember, we don't have this term now. Two x plus c minus X capacity to and what's the no doubt so here we don't have the prime. So as zero is warned, which means one plus C two is one. Then we know C two is still Yeah, we don't have to stern so actually ethics. It's one of the two e x plus e Quebec. This is the solution

All right. So we start off with F double prime is equal to the sign of X plus e to the power of two acts. So left prime is equal to the integral off f double time because they know where to go from. F Devil Prime Thio Prime two f. So we take after prime is equal to the integral off after prime and F is equal to you the role of F prime. So if we take the integral of sine of X plus diesel powered two acts, we get negative co sign of acts plus e to the power of two X to it in the two plus, of course, are constant. So now if we do f prime of zero is equal to what have so it's basically sold for our for our constant we have negative coast side of zero plus 1/2 eat the power of zero plus our constant it is equal to 1/2. So if we simplify this down, we have negative one plus 1/2 Plus constant is equal to half or C is equal to one so therefore we can write F crime is equal to negative co sigh of X plus 1/2 eat the negative two X, sir, That's eating the puzzle to X plus one. All right, so the next thing we're gonna do is we're gonna take the integral of all that. So to get f of X so out of X is basically equal to the integral of negative co side of X plus 1/2 e to a two x plus one to the ex could protect the interval of each of these things separately. We can get a negative sign of X plus e to the power of two X divided by four plus X plus our constant integration. So now we know that F zero is equal to 1/4 so we plug in zero and we set this equal to 1/4. But this is the sign of zero plus e to the power of zero or four plus zero plus are constant. So this is just equal to 1/4. Plus our constant is equal to what force, plus our constant. So basically your constant equals zero. So therefore we can write f of X f of X is equal to negative sign of acts plus e to the power of two acts divided by four plus X and that right there is your final answer

Okay. So you know that the derivative of the function we're looking for is one over two. Your checks plus pie. I'm sign of my ex. And so then you know why I was going to be an anti derivative of this function. So, thanks, minus co sign of my ex. Verify that's an anti derivative. And then plus some constant C. Well, we know that the graf passes it. Appoint one too. So why is equal to two when X is equal to one? Sits a squared one. Linus Co sign. Hi C two equals one minus kasan. Tobias Negative. One two is two one native minus thinking and one is two seeing to see zero. But that means our function. And C is hero. It's just read. X minus occurs on, Becks.

Okay, so we have a double Pine Cove X is equal to 1/2 each ex eastern and negative at a time. And Joe is he could be Oh, so you got one. You know, that time at the vaccine interval of that double time? But I think with two, 1/2 in and go of ex geeks and then they get attacked The X, which gives us 1/2 minus negative. But see, we know that a crime. And so oh, here is equal. Don't was equal to 1/2 this week in the native X or actually, these are through. Do we get that? He could go. So that's a crime. Could one have minus eaten in the ghetto? Now we know that f of X is equal to the integral of the first derivative that's integral of 1/2 e. T o minus. Eating NATO back the X, which gives me the same thing, would have needed access to these in the vexed. But he you know that girl you go to one because it would have defeated the girl plus 1/2 heated in girl. See? But it means that she is also he could sell. So our function at X is equal to 1/2 need to expose eats it and they're gonna back


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