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Question

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=x^{2}+3 e^{x}-2 x
$$

Prathan Jarupoonphol · Accepted Answer

I in this meal, we are solving the equation. Why primes Equal X square plus tree e to the eggs minus two X on this one is very easy because you don&#x27;t have to realize on integrating factors anymore because there&#x27;s no why there&#x27;s no Y term. Anywhere you have B y by the X equals sum function off eggs. This is entirely optics. You can use the technique hole separation, all variables, Which means you supreme eggs and why on to east Sigh off the equation and you can integrate them just like that. So here X square plus tree e to the X minus two eggs is a function off xolo. You can move this D x to the rice I and here we go. We can you degrade it both side. Since the wearables separate from each other, we gonna we&#x27;re gonna have. Why on the left Cy and some function off on the right side, which easily integrate the bowl. Uh, this is as cute over tree plus tree to the eggs, minus x square plus c. Now, if you cannot do these integration on the rice, I asked, as you go like this, you can always like practice. But these are really simple there, Polina Meal and Exponential with no further complication. So you should be able to do it with some practice. Also, do not forget the sea. And that isn&#x27;t. You can always shape if it true, by plucking Whiting. But that booby just reversing this integration. We just did. Right? So it is the correct answer. That is it. Nothing complicated. Thank you.

Taylor Shimono · Answer

So to find the general solution of our equation, I first decided to go ahead in rewrite. Why prime as D. Y t X and this is going to allow us to multiply both sides of our equation. Bye, d x, um, to find that d y is equal to three x plus e to the x d x. And now that we have a separation of variables, we can actually integrate each side of our equation here. So on the left side, we&#x27;re going to get the integral of one D y, which is just going to be why. And this is equal to the integral of three x plus e to the x t X. So we know that the integral of a three X is going to be three over two x squared, Um, plus the integral of each of the ex, which is just e to the X. And finally we can&#x27;t forget, plus our constancy. So we found our general solution to our equation by using a separation of variables and then

Prathan Jarupoonphol · Answer

question. We are fighting a general solution for this or the so we put it in standard form. We have that p of X is minus two So integrating factor is e to the minus two X and we must fly True, you&#x27;re gonna have Why times integrating factor off it prime equals three eggs e to the minus two eggs And so we want to integrate the rice I write to fire Why? Let&#x27;s do this. We pulled three ofus and use into gold bipod as always you will be x the we will be need to minus two x d eggs So our the you will be just the X and we gonna be minus the same function over too. Now we put it in you We will be ex Thanh the exponential or what to my last time minus will become plus integrated e to the minus two X over too d x And this is just exponential and we can duty integration right away. There&#x27;s going to be home minus e to minus two eggs over four. Yes, And don&#x27;t forget the car sent to So we have integrated the right hands. I it will be this times tree trying times e to all this. Um, go back here. We have that. Why? Times need Thio minus two. X equals minus tree X over too. Minus tree over full. This to turn times I need to. The mine has two eggs, plus constant o time. Also, Egypt in my nest to x. Sorry, there is no, There&#x27;s no e. There&#x27;s no function there. It&#x27;s just constant. Er Okay, so when we remove this, it&#x27;s gonna cue this one. But he&#x27;s gonna add e to the two X over there, and that is the answer. Let me rewrite so minus trio to x minus three or four plus c A cause, then time e power to its And that is Thea Answer. Thank you.

Prathan Jarupoonphol · Answer

in this video, we are solving the equation. Why equals? Why Tom&#x27;s X square plus one? And if you are familiar with the subject, you can see right away that we can use separation or variables this white times function of eggs with no like no further her and be hired. You can move I into one side eggs to one side. So we have this 10 y de vie de clothes is quiet plus one times the eggs. And of course, we gonna integrate. What&#x27;s I this Give Look off. Why the natural lock And this is just pulling a mere right. So x Q over three plus X plus scene just some constant. And so, if you want why And we have long why here we can use then relation off locker retune and exponential He had the best Sorry, the best can be any base. But now lock How in here would mean best is e right? It means be into the C equals to a So this dean is our see the best is E. We have whyy clothes e power by this this holding X cube over three plus x plus c and that is it? You can stop here. And it would be correct that sometimes you want to there with this constant for the you have e to the C tons into the function of eggs. Some some people prefer to write this ass another like constant entirely right as some some big C. And that&#x27;s why you will see this kind off answer, but is essentially the say is already is already done. By the time you hell this constant here. Just just to show you that is the same. This big c is just constant e to the C and that is it. This is the general solution. Thank you.

Ernest Castorena · Answer

ask us to give the general solution of given differential equation. I mean, those just take the indefinite integral of both sides. Well, it is integral of the right side. Look like Well, the top looks like that. Irritable in the input, right? Almost. It should be a negative drinks, except the negative two is gone. Well, why would that be? We have input to the three. Helps power. Well, to get that they were writing in the diary room. Negative. So that negative 1/2 what it got a friend and the negative two would have canceled out with that device it. So let&#x27;s see. Well, he would get ISS one over the square of one Linus expert. But what is this weight enough to get indefinitely? Royal Constant. If we take the independent in a row one more time to get original function. This right here is that it really do, uh, sine inverse of x. So the indefinite and I rode Ellis. It&#x27;s sine inverse effect. See, I lost another constant beat on. This is the most general form. No, I think they want this to you see One and c. T. Barnum&#x27;s. Okay. Mark skin

Smita Praharaj · Answer

in discussion there, Astra. Find the general institution off the differential equation. Why Prime equals to y O operates. No. What? Why is considered as a function of next? So white crime would be d y over dx. And that&#x27;s why you&#x27;re Rex If we bring y to the bottom here and take the d x to the tough So we get d y o ver y equals two d X over Rex. Now integrate both sides of this previous equation. So we get Ellen absolute value off. Why calls to Ellen Absolute value of X plus C where C is any constant. Now I can write. See as Ellen off a for some constant A So we get Ellen. Absolute value of y equals two Ellen Absolute value backs plus Ellen, eh? Now, by using laws of logarithms, this boils down to Ellen Absolute value of a X. So Ellen, absolute value off by equals to Ellen, absolute value of the X or Y equals to a X. So this is the general solution to this differential equation. Here. We can also check if the solution is correct or not. If why is they extend the Y over DX equals to a which is same as eggs over eggs. Now a X we solved for is why. So that&#x27;s why Rex and that&#x27;s precisely what under potential equations.

Problem

For the following problems, find the general solu…

Problem 266 Hard Difficulty

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=x^{2}+3 e^{x}-2 x
$$

Answer

$y(x)=\frac{1}{3} x^{3}+3 e^{x}-x^{2}+C$

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$y(x)=\frac{1}{3} x^{3}+3 e^{x}-x^{2}+C$

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Anna Marie V.

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Gilbert Strang,Calculus

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Gilbert Strang,

Calculus