verify-that-yxx-x1-is-a-solution-to-the-differential-equation-yfracd2-yd-x2fracd-yd-xfracx32-x2-31x3

Question

Verify that $y(x)=x /(x+1)$ is a solution to the differential equation
$$
y+\frac{d^{2} y}{d x^{2}}=\frac{d y}{d x}+\frac{x^{3}+2 x^{2}-3}{(1+x)^{3}}
$$

Michael Jacobsen · Accepted Answer

in this problem, we have a second order differential equation for which we want to determine if the falling function is a solution. Well, since the equation itself is of second order, the first thing we need to do is compute the first derivative D y T X, then the second derivative due to wide the X two. So first, that first derivative D Y DX can be done through the quotient rule. And with the quotient rule, we first take the derivative of the Numerator X, which will be one itself and multiply it by the denominator. So copy X plus one that&#x27;ll be minus the same kind of process. But in reverse order this time we copy the numerator X and multiply it by the drift of the denominator X plus one, which is just a one itself. And this will be all over the denominator squared or X plus one to the second power. Since we&#x27;re going have to do quite a bit of work with these derivatives, it&#x27;s a great idea to simplify these as much as possible. We have X plus one minus X itself left in the numerator, so the first rib have simplifies down to one divide by X plus one squared. Let&#x27;s go off to next. The second derivative, de Tu y DX two for the second derivative can be done in a number of ways. Let&#x27;s express the first derivative as follows. It is X plus one to the power of negative, too. We want to take the derivative of that result expressed in Newton&#x27;s prime notation. The result will be first, the drift of the outer function negative too Times X plus one to the power of negative three and through the chain rule, we take the derivative of the Inter Function X plus one. Luckily, in this case, that derivative is times one, and so we can simplify this entire results into negative to divide by X plus one to the power of three. Now we&#x27;re ready for the next step of this process. In order to verify that this function is a solution, we need to go back to the original differential equation and work in multiple steps. Let&#x27;s first try out the right hand side, all right, or a just for right hand side to see how that expression simplifies the right hand side of this differential equation is first Dwight D. X, but we have by substitution that D y t X is one over X plus one to the power of To there will be, plus the function X cubed plus two X squared minus three. We&#x27;ll divide by X plus one to the power of three and where to do a quick check on this? Whether it&#x27;s a solution to the difference, you&#x27;ll equation. We should simplify this answer as much as possible. It&#x27;s all come down to the first fraction. Multiply it by X plus one in the numerator and denominator, and the result will then be equal to X plus one plus X cubed plus two X squared minus three all over the common denominator of X plus one. Cute. So now this simplifies down to X cubed plus two x squared plus X minus two. All divide by X plus one to the power of three. And that&#x27;s our simplified form of the right hand side. Now let&#x27;s call this portion of the differential equation the left hand side. Our job is to do the following. We&#x27;re going to substitute as before, into the left hand side, and our goal is to try to get to this format here. If we can get to this, resulting expression after simplifying than a work will be complete. That indeed why of X equals X divide by X plus one is a solution. If we cannot get to that step, then we would know that this is not a solution. So it&#x27;s now start off again from the left hand side. The left hand side first involves the function. Why so by substitution? Why is X divide by X Plus one? They&#x27;ll be plus the second order derivative, which we simplify down to negative to divide by X plus one cubed so B plus negative, too. Divide by X plus one to the power three Recall. Their goal is to simplify this as much as possible. So let&#x27;s go to the denominator of the very first fraction and multiply it by X plus one the power of to and again X plus one to the power of to and the numerator of that fraction. Then the result will be X times X plus one squared minus two. All divide by X plus one to the power of three. So we&#x27;re nearly there. Let&#x27;s first work on expanding this expression. Here we get X times X squared plus two X plus one, then minus two. All divide by X plus one to the power of three. Now, once we distribute the variable X into the group, we obtain X cubed plus two X squared plus X then copied My is too all divide by X plus one to the power of three. Comparing both sides the right hand side with the left hand side we find that these expressions are equal So the left hand side is equal to the right hand side. And why of tea? Excuse me. Why of X, which is equal to X divide by X plus one is a solution, since these expressions are both equal.

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

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

Justin Avila · Answer

So in this equation, um, we&#x27;re going to I&#x27;d separate them in tow em, and and And since the taking the directive of each of them will is would have taken, like, a long time to sort of find out. Um, I kind of wrote it out here. And although they may look a little bit different, they&#x27;re actually exactly the same when you account for the negative one and simplify it. Because if we just do it, then still both of them will o m n n will equal why squared minus minus, x squared all over, um, X squared plus y squared and that entire thing square. But the this, uh, doesn&#x27;t matter. Like what? This is really. It&#x27;s just the fact that, like, they&#x27;re both exact, which is the main point. So getting into the main aspect of the lesson now and the equation we&#x27;re going to have, we&#x27;re now going to take the integral off each of them. And so for em, we&#x27;re going to. What we&#x27;re going to do is that we&#x27;re going to be taking its derivative. It&#x27;s integral by D x, and when we do sell, we&#x27;re going to get so for right Now we&#x27;re going to have extra negative first, which is going to be Ellen of Ellen of X. Then we&#x27;re going to have Then we&#x27;re going, particularly then we&#x27;re going to be taking the integral off this equation right here. And so there is a, um and so when taking this equation, there is a sort of a trick trick function that you can use where you can use arc 10. So I&#x27;m going to write down the side right here. But when taking the integral of arc tanne, um, not our camera. When you take the derivative of our 10 you will let&#x27;s say that we have the, um you take the derivative of our 10 X. What you will get is that you will get one over X squared plus, um, plus one. And so when we&#x27;re taking the integral of this since we&#x27;re gonna be dealing with why the equation will come out to be arc 10 times X over Why? And if you confused, like kind of why that is and how we really got to the sort of how you get this and how you get like all this, there&#x27;s you can look online to figure out like the whole like you substitution and like, the whole method as to why it is specifically X over. Why? But when you when you take the sort of, um But using like the whole trick are can rule and apply it to the integral, we will get this equation. And so when we apply this to our other, um, to when we apply this to our other equation where we have the end right here, what we&#x27;re going to have is that we&#x27;re going to have the integral of and whose we&#x27;re gonna have the integral of and by de y. And so when we take the integral similar to how we sort of did it in the M portion, the way that we&#x27;re going to have it for the end is where it&#x27;s going to be. The is where we&#x27;re going to actually have it, where we&#x27;re going to have it, where it&#x27;s going to be arc 10 times. And so since in this one we had it where it was X over y. Since this one is sort of the in a sense is going to be a different since this one is the opposite innocence where it&#x27;s instead, where this one was, Why this one&#x27;s gonna be axle Instead we&#x27;re and we&#x27;re treating this with respect to D Y. We&#x27;re going to have here is that we&#x27;re going to have why, over X and and so that the reason why we kind of have that one right there. So when we&#x27;re going so when we have the equation, we&#x27;re going to have it where C is equal to Ellen of X minus arc tanne of I just go right arc t Just for actually now just right are tan are 10 for X over. Why? And we&#x27;re gonna add that with Arc 10. And that&#x27;s going to be why over it&#x27;s going to be, why over X And I just messed up that right there. And so that&#x27;s what this equation is going to be

Sajin Shajee · Answer

the best function. Ensure that using the given equation extent liable five minutes. Why, prime? Uh, close x times y is equal is your training determine that Why one eat going X Times J one, uh, is actually part of is part of the equation. So you have this eye space or to do why one prime equal Teoh exchange gay zero x Why Devil Prime Fly one is equal K zero Next exchange one X salary drinking And don&#x27;t be accident X. Thanks. Queer off K one of tanks Buying is excellence, you zero weird zero. So it satisfies what I want Pickling extra one off is is a solution of equally.

Mostafa Hassan · Answer

I want this differential equation D y by d X equals y squared over a squared plus one. But concerning question and the question number one If we consider that why not take it with you? We can separate are variables by dividing boy, why square to get one over y squared Do you worry? Equals to one over x x squared s one e x Now we want the great port So get our solution We can solve its integration by substitution over a squared plus one the X This indication can be solved using substitution by substitute x for you then you questo x the integration bite dramatics Prostitution 1 to 10 teeth you coastal Then seat is in you. Engquist was take a square Sita. The seat is in view over plus B squared. It puts too. Take a square seater. The cheetah were one plus 10 squared seed because that six square seater a close to one less 10 square seat. Then six square seater. These heathens where one lost tennis square meter a cruise to from last tennis were Sita this heat over one 10 square? Sit. Forget medication, sir. Then we will hope. Take it off the zita a question seat. Reverse substitution toe Get U equals toe 10 seat, then Sita. Because toe 10 members you I said, do you overwhelm? Plus U squared a course to then in verse you will, plus c most destitute thing is this tradition to get the X well for from lost excess Quit. It was then and verse X. I&#x27;ll see. Are you going? Our question number two, We want to integrate with the left hand side. Then our solution needed to worry. It was a bar. Negative form. He goes to Dan in verse. Takes us tea. Then one over. Roy, It was negative in verse. Thanks. Negative seat was in. Why? If X equals two on minus 10 in verse. Thanks. No negative seat. This is a general solution to equation number. What? But you can&#x27;t that we want to solve the question number one before dividing by westwards. We consider that why not equipped with you? That our generous with our general solution toe question number one Is this a question and or too busy? Thanks, everyone

Problem

Verify that $y(x)=e^{x} \sin x$ is a solution to …

Problem 6 Medium Difficulty

Verify that $y(x)=x /(x+1)$ is a solution to the differential equation
$$
y+\frac{d^{2} y}{d x^{2}}=\frac{d y}{d x}+\frac{x^{3}+2 x^{2}-3}{(1+x)^{3}}
$$

Answer

$L H S=R H S$ $y(x)=\frac{x}{x+1}$ is a solution

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$L H S=R H S$ $y(x)=\frac{x}{x+1}$ is a solution

Differential Equations and Linear Algebra

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Integration Review - Intro

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Stephen W. Goode, Scott A. AnninDifferential Equations and Linear Algebra

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