Solve the differential equation and use a calculator to graph several members of the family of solutions. How does the solution curve change as $ C $ varies?
$ xy' = x^2 + 2y $
Okay, question given is X Y s is equals two X square plus two y and we need to find values off. Or we can say solution off this differential equation and we have to see with change off, See how the graph of this equation changes. Okay, so first, let's start with writing. This is X d y by D. X is equals toe X square plus two way. Okay, So if you see we have independent X function and why here and your ex? So let's just divide by X to see what we can do. So therefore, we'll have divide by D X s X plus, too. Bye bye, x. Okay, so we have your function. Where to buy excess multiplied by why and an independent x Okay, so it looks like d y by d. X plus p y equals two Q. Where p and Q are function off X only. Okay, so if you change a little bit, we'll have this form. So do you have I. D. X minus two by X y is equals two X and now it's completely look like the given equation. And in such kind off equation, we have directly are integrating factor as it is to integration off PD X. So we'll calculate are integrating factor he raised to integration off minus two by X DX. Okay, so you're minus two is constant. It will come out times integration one by X. Right. So one by axis. Lennox. So that's the answer for integrating factor. But we can simplify this if we take this minus two s power off eggs. So it is toe Ln X rays to minus two. Right. Since log M times log A is logged a raise to em. Okay, way have. And now we can apply another property. Here we have e and Ellen has based e. So this becomes express to minus two. What's the property? It is to log m to the base A is m. Okay, so we have our integrating factor now. So now solution is solution is what White times integrating factor is equals to integration off few times integrating factor d x plus c. So why times integrating factories? X rays to minus two is equals two Q is how much Q is eggs so X times integrating factories. X is two minus two d X bless see. So why upon X square, right? Exodus to minus two is nothing but one upon X square. Similarly year x upon x square The X plus e So you can cancel one off the X So we'll have a solution is why over X square is equals to what is one upon X integration one upon X Integration is l Lennox right? Or you can remember it in reverse Order that different station off Lennox is one over X. So this is a Lennox. Let's see. So this is a solution. Okay, Now we have to see how this graph will look like toe check. The graph will substitute some of the values and see okay, or you can use calculator graphing calculators so that these were X axis. Why exists for X equals to zero? If you see this becomes infinity infinity right? And let's let's assault force equals to zero first. So at X equals to zero function is why Why over zero and 11 0 is minus infinity. So this is infinity minus infinity. So we don't know. It will be something at infinity at X equals toe one. This is why let me calculate this year at X equals to one. We have y over one. Ellen off. One is zero and C zero. So why Paul? So zero. So we have at X equals to one wise zero. Okay? And doing the same process. 42 poor points. We'll have a graphics. Something like this. Okay, Now, if you increase this, let's say we take x CS two CS three and we calculate so we'll have a graph like this for Let's say this is zero. So this is C one, This is C minus one C minus two, and similarly here. Okay, so that's a graph for differences. Okay. Thank you.