for-the-following-problems-find-the-solution-to-the-initial-value-problem-yprime3-y-x6-x2-y0-1

Question

For the following problems, find the solution to the initial value problem.
$$
y^{\prime}=3 y-x+6 x^{2}, y(0)=-1
$$

Robert Daugherty · Accepted Answer

so Ford up five problem to 77 were asked to solve this first order linear, ordinary, different show equation. So the key to solving this will be to get it in standard form. So why is equal to back up? Why prime plus a function of X times y is equal to another function of X and then my integrating factor will be e to the anti derivative of P of X. So standard form for this guy&#x27;s gonna be y prime minus three y is equal to minus X plus six x squared. So the integrating factor is going to be e to the minus three the X So this is going to be e to the minus three X. So this equation is going to be why he to the three acts Prime is equal to minus X e to the three x plus six x squared e to the three X and sorry about that. It&#x27;s minus. So it&#x27;s minus three x and minus three X was the integrating factor there. So now the key will be, then, just to integrate both sides of this equation. So on the left side, you&#x27;ll have why e to the minus three x and then when you perform this integration on the right side, minus 1/3 you to the minus three x and you&#x27;re left with this polynomial terms six x square plus three x plus one, plus a constant integration. So then to solve for why, why is going to be minus 1/3 six X squared plus three x plus one plus c e to the minus three x Excuse me plus three x when you multiply that by negative 1/3 6 X square plus three x plus one ce to the three X that we have an initial condition y of zero is negative one. So if y zero equals negative one, then I get negative. One is equal to minus 1/3 zero plus zero plus one plus c and then e to the zeros just simply one. So this is minus one equal minus 1/3 plus see therefore C is equal to minus 2/3. So this final answer becomes why equal minus 1/3 six x squared plus three x plus one minus 2/3 e to them plus three x And that&#x27;s the final answer

Nick Johnson · Answer

Okay, so we have the different equation here and why Double crime plus six x is equal to zero. And then we have that. Why of zero? Basically one. And then why? Prime of zero is equal to two. So we can go ahead. And we can rewrite this difference equation as why step a prime is equal to negative six acts. Okay. And then we integrate both sides twice. So the first time right? Integrating here we get that or the growth of why it double crime is the first derivative. My prime is equal to well in a girl here and give us a negative three x squared plus adultery constant. It&#x27;s called Seat one. Hated Integrating again. Gives us why is equal to, um, negative X cube. Um, right over 300 canceled. Just get negative X cube. And then we get Well, um, in agro of C one is going to give us plus C one times X. We pick up another constant, uh, see too. Okay, so now we go ahead and reuse our given initial conditions. So we know that why prime of zero equals two. So why primes? Zero. Um, is equal to to give. That implies that two is equal to see one. You know that why of zero is equal to one came off. That implies that one is equal. Ju si two. So therefore we get, um, a solution to our initial value problem. So our solution here is why Ivax is equal. Chew um negative X cube. Plus she wrecks plus one her it

Prathan Jarupoonphol · Answer

this video. We are going to solve this equation. Uh, x times the wide body X equals y times X minus two. And you can see that this is one off the cases that we can use. Separation off, very boat. We can move. Why do the lep, sy and X to the rice? I So we have this wanna why d y equals one minus two over x the ex right? Because we divide by eggs. Now we can just integrate both sides, right? So long, Absolute. Why equals, uh, eggs minus two long absolutely x plus some constancy. Now we can Well, this has lock off X square. Right by the I did a D off locker room. And absolutely Looz choir is just just like pulling a meal. Right? And Ah, don&#x27;t worry too much about this absolute value off. Why? For now? Now we want to get rid off Mark. We want just why, right? So we use the right hands, I power it, Miss e, We&#x27;re gonna get this thing so X will become e to the X and the miners. Long X square. Gonna be X square below here, into the sea. We can rename it as just some big C is also a constant. We want to fi what it is. So we put the battery condition in. Why is tree when egg is one? So B times C equals tree. That means our big C is three times e to the minus one. Ah, since is just Rio number as well. So this is the whole thing is a number. So don&#x27;t don&#x27;t be confused with with e to the X. Now we pluck this in. You&#x27;re gonna get, uh, tree e to the X minus one over X square, and that is the answer. This one is a bit unusual because the constant, um, multiply their function. Not not ab to the function. So you don&#x27;t have this like you don&#x27;t have the last term as a constant it embedded in the function. Thank you.

Alexandra Embry · Answer

in this problem were given a differential equation over here, and we want to find a solution for it that satisfies these initial values that are given. So to do that, we can go ahead and in a great to get that Why double prime of X is the indefinite integral of six x dx, which is equal Teoh um one Halftime. Six So three x two The two plus C one. Um, that&#x27;s just using the reverse power role here. So then we can go ahead and integrate both sides again to get why prime of X equal to the integral of three x squared plus C one d x, which is equal Teoh again using the power rule, um will be X to the third plus C one x plus some constant c two And finally, we can integrate one more time to get our solution. Why have x equal to the integral of x? 1/3 plus c one x plus seat to d X equals 1/4 x to the fore plus 1/2 see one x squared plus c two x plus, some constant C three. All right, so now we have all that and we want Teoh plug in these initial values. So this is a general solution to our differential equation. And now we want our solution plugging in our initial values. So we&#x27;ll start up top with this wide, double prime equal to three x squared plus e one. And we can plug in our condition over here that we want. Why? Double prime of zero equal to four. So why Double prime X equals three x squared plus C one. So we want four equal to three times zero squared plus C one, which means see one is equal to four. All right, we can use this in our second equation. Will look at why prime now and our condition on why Prime? So we&#x27;ll do that right here. We want why Prime of X is equal to X to the third plus C one of X. Where do you know what C one is? It&#x27;s four. So we can just write four x plus c two and then we want to plug in our conditions. So why prime of zero? We want to be negative one equal to zero to the third, plus four times zero plus C two so this gives us that si two is equal to negative one and then finally will use this condition that we want y of zero to equal one supposed to be in blue. Why zero equal toe one. So we can use our equation for why here and plug it in. So we have rat Why the vax is equal to 1/4 x to the fourth plus 1/2 See, one is for we know X squared. Plus Now we know C two is negative one so negative one times x plus C three and we want I have zero equals one. So we set one equal to 1/4 0 to the fore, plus 1/2 times four times zero squared plus negative one time zero plus C three All the zeros cancel and we get that one is equal to C three So we can finally plug that in Teoh r Y of X to get a final, um, solution to our problem. So our final solution will be why vax equals 1/4 x to the fourth plus half of fours to two x squared minus X plus one. And this is a solution that satisfies all of our initial conditions and solves our differential equation

Bobby Barnes · Answer

we want to solve this initial value problem here. So what this really saying is, we want to find some function. Why? When we take the second derivative of it, we get to minus six X. And when we plug in zero into its first derivative, we get more. And where we plug it zero into itself, we get one. So let&#x27;s just go ahead and first take the anti derivative of this. We could figure out what our general solution for the derivative should look like. So I was gonna rewrite. This is why Prime Cause If we enter it the second derivative, we get the first derivative. Why prime? And then we go ahead and distribute our integral here by the linearity of it. To get interval of two, he x minus six times central of X and now integrating Each of these will integrate a constant. We just multiply by the variable that were integrating with respect to we get to x minus and then this will be six x power rule says add one to the power, then divide by. It&#x27;s do power and we add a constant C. So we&#x27;ll go ahead, simplify this down to two X minus three. Ex spared lost seats. So this is why Prime and we know when we plug in zero, you want to get it out. Put up four. Let&#x27;s go out and plug that in. So we&#x27;ll do this in green down here, So we&#x27;re gonna have or is equal to Well, zero minus zero plus c. So that just gives us four is equal to see. So I&#x27;m gonna go ahead and move this up here then. So we know now that why prime is equal to two X minus three X squared plus four. Now, if we want to get why, we can go ahead and again integrate each of these with respect, X So in the interest of DX integral DX and again, bogo had implied the linearity of this to get two times integral of X d x minus three times and a roll of X squared DX plus the integral of four. And we&#x27;ll go ahead and use power rule to integrate this again. So we&#x27;ll have to be X squared over two again. And then this next one so have three times x Q two raced. You&#x27;re excused. Divided by three and then again, constant. So we just multiplied by X. And then we have C so it&#x27;s going to simplify this down a little bit and this is gonna give us that. Why is he going to X squared? Minus. So those threes cancel X cubed plus four x plus c. And again, we have that when we plug in zero, we should get out one, so it&#x27;s gonna be one is equal to zero minus zero plus zero plus c. So that just says, See, is one and maybe over here I should say, this is C one and this over here is C two just so there&#x27;s no confusion off these being the same or not, because these are different variables so that we could just go ahead and plug these two things together. Now let me do this. And black, we got blood, but both of those together, and that&#x27;s gonna give us that. Our solution is why is eager to x squared minus X cubed plus or X plus one

Bobby Barnes · Answer

we want to solve this initial value problem here. So this is just a fancy way of saying find some function. Why such that when we take the third derivative of it, with respect X, we can get six. And we also have these other three conditions that are satisfied when we plug zero into the second first, an original function for a second and first derivative and our original function. So let&#x27;s first go ahead and integrate this first with respect to X to get our second derivative. So I&#x27;m just gonna write The second derivative is why double prime. So integrating a constant. We know we would just multiply by whatever we&#x27;re integrating with respect to which is going to be six X and then we&#x27;re gonna have some constantly can&#x27;t see one that we have to add to this. Now we know when we plug zero into this, we should get out negative eight. So let&#x27;s go ahead and do that. We&#x27;ll have negative eight is equal to, so plug it and zero will give us zero plus C one, and that&#x27;s gonna tell us that C one is just equal to negative eight. So we go ahead and plug that ends we have why Double prime is equal to six x minus eat. So let&#x27;s go ahead and integrate this so we can get our first derivative. So again we&#x27;re gonna enter each side with respect to X. And I&#x27;m just gonna go ahead and do that up here so we should end it with the first derivative on the left hand side and on the right hand side. Well, we can go ahead and use the linearity of Integral to distribute this across thesis traction as well as pull out that six so that we can write this as six times the integral affects minus the integral of eight. Now, to integrate X rays power room, it just has one for its power. So it&#x27;s gonna beat X to the first, and then we add one to the power and then divide by its new power. And then over here again, we just multiply by the constant or by the variable that were integrate would respect you, and then we&#x27;ll have a new constant seat. So it&#x27;s got a simple like this. So this should be three x spared minus eight x plus C two So this is why Prime and again when we plug in zero, we want to get out. Zero. So it&#x27;s going to pluck that in, So be zero minus zero plus see, too Well, that just tells us that zero was eager to see to. So we can actually just come up here and cross out that C two. So that&#x27;s gonna be y prime. And then lastly, we can go ahead and integrate this again with respect to X, And we could go ahead and do that up here. So that&#x27;s gonna give us why is equal to and we&#x27;ll use the linearity once more to rewrite this as Thean Toral or three times intro of X squared minus the integral eight times the integral of X. Now, to integrate, each of these again will use power rule. So it&#x27;s gonna be three x squared. Well, we add one to the power and then divide by the new power and that over here again, add one to the power divide by the new power and then we have our third constant C three. So simplifying this down, which I have execute minus four x squared plus C three so this is he could&#x27;ve why? And we can go ahead and use the fact that when we plug in zero, we want output five. So be five is equal to zero minus zero plus C three That just tells us five is equal to see three. So now if we go ahead, combine these two together, we can get that. Our solution is why is he going to execute minus for X word, plus by. So, if you were to go ahead, take the derivatives three time, we should end up with just six as well as those other initial values as well.

Problem

For the following problems, draw the directional …

Problem 277 Hard Difficulty

For the following problems, find the solution to the initial value problem.
$$
y^{\prime}=3 y-x+6 x^{2}, y(0)=-1
$$

Answer

$y(x)=-2 x^{2}-x-\frac{1}{3}-\frac{2}{3} c^{3 x}$

Related Courses

Calculus

Related Topics

Discussion

Top Calculus 2 / BC Educators

Catherine R.

Anna Marie V.

Caleb E.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

Gilbert Strang,

Calculus

Catherine R.

Anna Marie V.

Caleb E.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$y(x)=-2 x^{2}-x-\frac{1}{3}-\frac{2}{3} c^{3 x}$

Calculus

Catherine R.

Anna Marie V.

Caleb E.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Gilbert Strang,Calculus

Catherine R.

Anna Marie V.

Caleb E.

Joseph L.

Gilbert Strang,

Calculus