solve-the-given-differential-equations-d-y3-y-d-xe-3-x-d-x-2

Question

Solve the given differential equations.
$$d y+3 y d x=e^{-3 x} d x$$

Bobby Barnes · Accepted Answer

we want to solve this differential equation here. So notice first, This is a separable equation because if we multiply each side by eat the wind, he can&#x27;t sell it on the right and on the left will have e to the wife Times do I buy D X is equal to three x squared Now let&#x27;s go ahead and multiply O r dx over and I say multiply. It won&#x27;t. It&#x27;s because we can&#x27;t really multiply by t x Uh, but now we&#x27;re going to integrate each side. Now the inter grody to the why should just be eat. Why the integral of three ex cute Well, that&#x27;s gonna be power also should be excused and then we divide by three So the threes cancel and then we have our constant C. Now we just need to take natural log on each side so we can get why by itself. So why is going to be natural? Aga X Cube? We&#x27;ll see. So this here will be our solution to this differential equation

Nick Johnson · Answer

Okay, let&#x27;s consider the differential equation. Um, why? Minus X d y The X is equal to three minus two X squared de y dx. Okay, so there are some derivatives. We have an equation. So we have a differential equation. Okay. So we can rewrite this equation as well. Why minus three is equal to acts. B Y the X uh, minus two x squared de y the x. Okay. And we get here. Um, let&#x27;s see, what we get is, um Well, why minus three is equal chew. You can factor out and well and x minus two x squared times. Do y dx. Okay, um, so we separate the very rose, and we integrate both sides. So, um Well, what we&#x27;re gonna do is we&#x27;re gonna integrate, um, one over X minus two X squared D X is equal to the integral of one over. Why minus three. You want? Okay, So how we do this in a grow? Well, we could use partial fraction decomposition, So we first do the partial fraction be competition here. Um, quickly. Go through this right. We have Well, um, we have one over X minus two. X squared is equal to while a over our for effective ex. Um, plus B over one minus two x. Okay, so we get the one over X minus Direct squared. Well, this is equal two looks is equal. Teoh. Um, well, a times, right. Get a common denominator. Eight times one minus two x plus B x over. Um, X minus two x squared. Um, and since right that I mentor equals that the numerator equal and we acquaint the coefficients and we get that a you find out from find that a is equal to one. Um, and the negative to a plus B is equal to zero. So therefore, B is equal to two. So, um, we find that our partial fraction decomposition is, well, one over X plus 2/1 minus x. So we could that integrate. We can then integrate. If we do the partial fraction the competition, we can integrate one over. It&#x27;s the X minus two x squared. The X is equal to the integral. Uh um one over. Why? Minus three. Do you want? Okay, um, so that the integral of well of one over x plus two over one minus two x d x is equal to well, the integral off one over. Why minus three. Do you? Why, um, so we split these up as the integral of one over X? Um, yes, we put these up, and what we get here is Well, um, the natural lawman off the absolute value of acts. Um, minus the natural log hopes that natural minus the natural log of the absolute value of two X mites. One plus the natural log of C one is equal to the natural log of the absolute value of why minus three. Okay. Um so, Well, we get here the natural. So that&#x27;s where we&#x27;re at. Right? Weaken. We could, um, combined this right Based using parts of large and saying that natural log of C one X over two X minus one is equal to the natural log of the absolute value of Y. Minus three. Okay. Um, well, you know, they think exponents of both sides, um, and get well e to the natural log of C support. X over two X minus one equals e to the nets. A lot of, uh, Y minus three just gives us, um, see someone X cease of one x over two X minus one is equal to well, y minus two. Okay, so therefore, why? Why? Right is equal to a cease of one X, um, over two X minus one plus three. Rightly and the three here. So there&#x27;s why. Um but then further simplifying and then simplify this out. Um, by factoring. And what we end up with is, um Why is equal chew? Uh, see? Ex minus three over two X minus one. You have it. All right, take care.

Banhishikha Sinha · Answer

So we&#x27;re given three X Mine stool. Why? Okay. Why? By the X physical Stroll three. Why the chickens ride? Streamline stool. Why buy? Do you want a blind? The X this cycle Stool three. Why? And by X market does one. Now I&#x27;ll speak to you again. You being physical school. Why buy eggs? Why is a little mean sex more after this intruder? The wine. My ds was still me. Thanks. You mean why the Knicks? So now they question one No decoration. One becomes train minus to B. You know me X TV by the X God still dream me. So be blessed eggs. You be by the year. Is it close to tree V by training? Linus, don&#x27;t we kiss this one now? Taking me on the right side? The God do, by the way, is a coastal three B by three minus to B minus. B x B by D. Eggs is equipped stool. No taking the sm three B minus. Be tree line. Don&#x27;t be. That is a coastal three B minus through you&#x27;ll be square divided by three minus. Don&#x27;t be so that can filled So x B by the X is a Constable, you will be square my nose. Three. Still see. So this equation is a separate The differential equation. Hands three minus to be divided by toe B squared Daisy. Is this still be X by TV integrating the left side by Be in the right side by X we get three minus. You&#x27;ll be by two Will be square TV musical stool. When by a the x no were gonna separate. Do Ah, school might be square baby minus one by the squid. Davey is a crystal. The eggs, my ex. This becomes minus three by go one by B minus. Met your log. Oh, which is a good school, Ellen. It is not your lawful uh, no. See, I do. No taking this this on the right side. And then No, Sorry, taking this. Keeping this here and this on the right side. You get well in a metre, long off eggs, she&#x27;ll be cause to minus C minus. Tree my toe one by me. So there&#x27;s a constant entry log X b. It is still minus C minus three By to when I where C is in arbitrary, constant against substituting. Did you do? Why? Cause V X You get natural law or why This is not your love of why. Yeah, growth to minus saying when? This three by still. And this becomes one by one by, uh one by why by X with your cones Local Why was minus trees? I do the course. Explain away. Why? Because to see you names every it&#x27;s way. I see where See as it was for you to the ball of my C.

Samuel Hannah · Answer

So this problem we&#x27;ve been given differential equation, whichever rewritten on the left corner of the screen by dividing both sides by ex sweat. So this looks like a homogeneous differential equation. A little check this in the blue box by making the substitution t x for X and T y For Why so be right hand side function becomes T squared y squared plus free T squared X y plus t squared X squared and then this is all over T squared X word. Now in all the terms of the numerator and the denominator for is a common key squared factor so we can divide us out and we are simply left with the original right hand side function. So this is a homogeneous differential equation. This means that we can make the variable substitution b equals Y over X, which can also be written. That&#x27;s why equals v. Times X. Now we&#x27;re looking for D Y the x Onda. We confined this in terms of V by using that the product rule which gives us that d want the X is equal to B plus D V, the X Times X, and this will be substituted into the left hand side of differential equation of the right hand side. We will simply substitute in Why equals three times X. During this substitution, we get the V plus X DVD X is equal to B squared X squared plus free the X squared plus X squared, all divided by X squared. So again there is a common factor of X squared all the numerator and denominator terms so this could be canceled out, which leaves us with b squared plus free. The US want off the right hand side. Now for our differential equation, we notice that we can subtract V from both the left hand and right hand sides. And so our differential equation now reads ex DVD X is equal to be sward plus to be plus one on This is in fact a perfect square because this is equal to the plus one squared. So we know to separate the variables by bringing the V variables to the left hand side and the ex variables that right hand side in this gives us that one over the plus one. Squared TV is equal to one over x dx on to get a solution for V we will integrate both sides. Now the integration head is fairly simple with the integral of one of the V Plus one squared becoming minus one of that B plus Well, Andi, the integral of one over X becoming the natural log the next and we will aunt the integration constant. See that now we want toe solve this for V. So we need to rearrange this equation for the and this gives us that, um, the plus one is equal to minus one over Ln x plus e and rearranging. By bringing the one over to the right hand side, received V is equal to minus one plus one over Ln x plus e and that is all solution for they now. We want the solution formally, but we can use our variable transformation by the fact that why equals three times X. So our solution for why is simply why equals minus X times one plus one of the natural log of X plus C

Mostafa Hassan · Answer

now we want the differential equation you want by the X equals two X multiplying boy. Why minus one over its weird plastic. Now we can start by spiriting are variables to get 1/1 minus one The all it calls toe two x over X squared plus three d x Then integrate both sides. You can start division of this boat two weeks over X squared those three X Because all of this integral by substitution we can service to cute you boy x squared plus three then do you The eggs It was 22 x and the ex equals two You over two x an hour and a girl became two weeks. Where were you multiple oId boy? Do you over to X My really sorry. We will get Do you know one over you? Do you equals who then? Absolute. You just see bird lovers. Our substitution will have then x squared plus three lost seats And we can solve our second a bigger 1/1 minus one. Do I buy substitution or we also destitute? You falling to minus one. Then do you equals toe deroy And our dignity came one over you do you equals food in U plus C equals then, boy, I minus one plus c absoluto life one trustee Then our equation became then absolute one minus one equals golden absolutely takes a squared plus three plus I see by exponentially both sides will have. Why am I in this one? Closed toe Exponential in absolute excess work three plus c They will start separate our left hand side Extra attention Lynn X X squared plus three not the blood E power seat We will substitute to see one boy E It was a power seat is in our question. Became absolute wild minus one Because you see one at the blood boy Absolute X squared plus three in our situation Why X equals toe See one about the blood by absolute exist word plus three plus one. When boy graters ends, you and C one negatives you on that bloody boy X squared plus three plus one When oy is smaller, zero body be calling our first step. We divided by while u minus one To solve our definition equation and the solution obtained assume why not take a to Z or however as way We have already noted that why X equals zero then our generous solution consisted off the east to question thanks to everyone

Problem

Solve the given differential equations. $$d y+2 y…

Problem 4 Easy Difficulty

Solve the given differential equations.
$$d y+3 y d x=e^{-3 x} d x$$

Answer

$\frac{x}{e^{3 x}}+\frac{c}{e^{3 x}}$

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$\frac{x}{e^{3 x}}+\frac{c}{e^{3 x}}$

Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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