Solve the following differential equations dy ~er" ~Iny2 (V) ty' + 2y = (2 _ (+ 1, with y(1) = {_ (c) 16y 24y' + 9y = 0...

Solve the differential equation.
$ \frac {dy}{dx} = 3x^2y^2 $

Hello. We have to solve the given differential equation that is G. T. Y. Please develop breast away. It cost a hero. So. Exalted form of the seclusion and mr plus M press to cost +20 So the complexity of this will be minus B. Plus money. Scuttled off one monies for into into one upon to. So this will because two miners of one x 2 plus minus. You squirreled off 78 upon 12 ι.. So alpha is one way to beat is square root of seven. So the solution can bitterness into the power of Halifax Stephen cause she went side we tax plus you do cause be text. This is the solution. So this will be cause to the to the power minus X. Y. Do seven off sign Beat. ISAT scores out of seven x 2 of X. Plus. She too calls scurried off seven x 2 or Fax. So this is the and so I hope you understood.

Already. So we are continuing to solve some of these home of jeans differential equations using the uh really area equation method. And so for this one we are taking the second derivative of why plus two times the first derivative of y plus two times Y equals zero. And our auxiliary equation then is r squared plus to our Plus two equals zero. So for this one, I'm going to use the quadratic equation to solve it because I don't see any simple roots jumping out at me. So we'll take um just a record. If you have an equation X squared plus bx plus C equals zero, then our quadratic formula is negative B plus or minus the square root of B squared minus four times a times C. All over two times a. I'm going to apply that to this auxiliary equation and we're going to get negative two plus or minus the square root Be square ends up being four And we have -4 times a which is one times 2 -8. All over two times a which is just two times one. Okay, we're going to simplify that. This is gonna be negative two plus or minus the square root of negative 4/2. So we can pull the four out of there. Right? Um descriptive for being too. So we get negative to put their minus two times x squared is negative one Over two. So the two are gonna divide out real nicely and we are left with negative one plus or minus I. Okay, so these are our roots here, R one and R two. And so when we have complex roots are general equation are general solution take a little bit of a different form. So if our roots look like alpha, some real value plus beta times I. And our general solution is going to be some constancy one times E to the Alpha times X times co sign better. X plus a second, constant times E f X times sine of the index's when you use that formula and apply it to our specific routes here. And that will give us a general solution that looks like Y is equal to C one times E. In our case, alphas negative ones are gonna have negative X times co sign. In our case beta is equal to one, so co sign of X Plus AC two times eat the negative x sine of X. And we have no initial conditions to consider. So this problem is done and this is our general solution.

Hello. We have to solve the given differential equation. That is two of the survey monastery device. Man is very close to zero. So we can add the actual education for this to end this car Ministry. M Man. It's funny quest to zero. This is the use of the equation. Okay, so we can solve it to m squared minus off today minus 12 1. So root of this object. Education will be three plus the route 17 x four. Mhm Yeah. Yeah. M two will be three manages current of 17 upon four. Mhm Okay. Yeah. Okay. So we can write down solution for this given differential equation. Okay, so this will be cause to white cause to see even into the power and Monex plus you took you to the Pavlov mm two X. So when it comes to seven into the Pavlov three plus the square root of 17 upon four or fax AC two into it. To the power of three months is square root of £17.4 of X. So why will be constant into the power three by four is three X of I four is common. So this will be into the power route 17 upon seven into a. To the power of 17 upon four of X. Please sit into Italy power minus office to carry out of 17 upon full. It affects. So this is that answer. I hope you understood. Thank you.

Hello. We have to solve that given different selection that is to do to y minus of treaty way- of by close to zero. Mhm. So we can write it auxiliary question for this. Yeah so that is two M squared. Monastery m minus one equals to zero. So the roots will be americans too Monastery monastery plus minus the square root of B squared. That is monasteries where that is nine min 4 and 2 -1 and 2 to upon go into. So this will be cost 2 3 plus minus the square root of nine minus that is eight. Okay that is eight upon for So this will be cause to three plus square root of 17 upon. For these are the rules and monies three wife my four Plus the Square Root of 17 of 1 4. An empty will be 354- of the Square Root of 79 14. So the solution of the differential equation can witness by cause to save and into the power and Monex prosciutto Into the power of M. two x. Why call to seven into the power of German yemenis three x 4 plus the square root of 17 upon four facts plus sito U. To the power of em to that is three x 4 months. A squirrel of 79.4 into X. Okay so we will take comin into the power three X. Up on four. So this will be close to 7. 8 to the power of the square root of 17 upon four of X. Placido into the power of the square root of 17- of school out of 17 upon four of X. So this is the answer. I hope you understood. Thank you. This is the solution of the defense Delectation. I hope you understood. Thank you.