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Solve the initial value problem_(X+8y7x)ak+ (eyx? _ sec 2y)dy=0,Y(I)=IThe solution is (Type an equation using and as the variables. Type an implicit solution. Type ...

Question

Solve the initial value problem_(X+8y7x)ak+ (eyx? _ sec 2y)dy=0,Y(I)=IThe solution is (Type an equation using and as the variables. Type an implicit solution. Type an exact answer in terms ol _.)

Solve the initial value problem_ (X+8y7x)ak+ (eyx? _ sec 2y)dy=0,Y(I)=I The solution is (Type an equation using and as the variables. Type an implicit solution. Type an exact answer in terms ol _.)



Answers

Solve the given initial-value problem. $$y^{\prime \prime}-4 y=2 t e^{t}, \quad y(0)=0, \quad y^{\prime}(0)=0$$.

Given this differential equation, as was an initial condition. So our goal here is to solve this and we're gonna get some expression and see if we can plug this central position back in to find the exact value of C of and therefore a more exact answer. So here we have a separate bill differential equation on. We have two different variables. We have y and teeth. And so when we have that, we have to get the Duke two variables on opposite sides of the equal sign. So an easy way to do this here is just to multiply by DT. So if we do that, we get that deal, why is equal to you? Need to the negative t dt Um, and to know already take the integral of both sides. So this side is really easy to evaluate. That's just why. Um, but here we have to use u substitution because we have a negative t right here. So let's set in it. If negative t is equal to you, take the truth. Both sides we get that do you is equal to negative DT, which means that negative do you is equal to DT. So Now we can plug in you and we get that the negative integral of eat of the U do you is equivalent to this right here. Do not we gonna find this integral? We have. Why is equal to negative E two? We can plug back in for you right away. So negative t So we solved this differential equation. And now let's plug in our initial condition. We have zero is equal to negative E to zero power. Let's see. Um, so zero is equal to negative one her see, So C is equal to one. Um, is now we can plug this back in to our solution. So we get that. Why is equal to negative e to the negative t plus one, and this is our final answer.

Hello. So the question is taken from the differential equation. So given the differential equation, let me write to test by prime minus Y is equal to to t It to the power 20. And the boundary condition given S.Y at X X is equal to zero is equal to one. Okay, so this is first order differential equation, how we can solve it. It is like from DV over DT minus Y is equal to 20 into the power duty. Initially we can evaluate the integrated factor for this equation. So integrated factor, we can write it as if we compare this situation with, do you buy your word ET plus P of Y is equal to Q. Then this p integration p O D T is basically the integrated tractor. So let us evaluated ps minus one minus of DT. So that will be it to the power minus two. So, integrated factory Z to the power minus T. So let us multiply the C to the power minus T throughout the equation. So we get eat to the power minus T. DV over DT minus E. To the power minus T into Y. Is equal to duty into the power to t minus T. So that will be. So from there we get B. This is like the off E. To the power minus city into Y. And on the right hand side we get duty duty to t. Sorry booty eat to the power T. Into pretty. So that does integrate on both sides. We get exponential minus team to Y is equal to who is just a constant. Initially the take tears of was functioning into the poverty as second function then first function as it is integration by parts. So the first function as it is the integration of second parties into the power minus T. Into the poverty minus of the differentiation of first function that is one. And integration of second function that is into the poverty beauty plus he sees just a constraint of integration. So that will be E. To the power minus T. Y. Is equal to two into E. It to the t minus two DT plus. See, okay now we can take this E. To the minus T. To the I can say let us first evaluate the value of this constancy. So in order to evaluation given that by at zero is equal to one. So let us substitute T. Is equal to zero to our decoration. We get exponential to the power, zero is one and why it acts is equal to zero is equal to one. So that is equal to easier than hotel museo and minus one plus. So this implies C. Is equal to treat. So let us substitute this value of C. Immigration. We get E to the minus T. Into Y. Is equal to do T. E to D. These minus E. To the T plus three. So let us take this exponential minus two to the right hand side. So we get Y is equal to two and two B minus one Plus three into into the poverty which is the required solutions of the equation. So all these clears your doubt. This exponential minus even take to the item side will get. So that will be exponential duty. Okay? So you for that mistake So please check first power. So hold this clears your doubt and.

In the problem we have the way upon DT that equals 0.5 into Y -200. So it is develop on Why -200 That it was 0.5 in detail. So it is LNY -200. It is equal to 0.5 Deep blessing. Or Y -200 equals It about 0.5 plus c. This is blessing. So Y f t equals 200. Last eight powered 0.5 G blessing. So here We have zero equal 200 Plus eight power see 50 equal 200 plus eight parsi Already is -150 equals eight parsi. So Ln -150 equal see And YFT Equal 200. The last eight power 0.5 G Plus Ln -150. So this is equal to 200 Plus you power 0.5 G Into It. Bar Ln -150. Therefore it is written as y f t equal 200 plus eight power 0.5 G Into Oneness 150. Or we have to become -158 power 0.5 G Plus 200. So this is the answer.

Hello. So the question here belongs to the first order differential equations. So given that why climb plus two Y. Is equal to team two. To the power minus duty with initial value of Y. Y. X. Is equal to one is equal to zero. Okay so how we can solve the situation The situation resembled with our equation devoured 80 plus B. Y is equal to you. So in this kind of situation how we can solve this by evaluating the integrated factors. Integrated factor is exponential T into a DT. If the derivative factory state. So we get E. The value of pain cases too two Entity DT. So that will be exponential community. Okay now it is time to multiply the C to the power to do two out of the equation. We get It to the 20 day by over DT plus two Y is equal to B Into the -2 T. Into the duty. So we can write this as the over DT into why into E. to the 20 is equal to take so taking T to the right hand side and integrating, we get D O Y E to the treaty is a curtain T. D T plus C sees the constant of integration. So here we get Y O E. To the duty is equal to please feel by a group Plus one plus let us use them boundary condition which is vie, it X is equal to one. Y. It is equal to one is equal to zero, substituting T is equal to one. We get vie sorry, the value variety is equal to one Is you? So that is you And this will be T is equal to 1. 1 x two Plus. This implies sees equal to -1. Let us substitute this value of seeing equation. We get we get by E to the T. Is equal to the square right oh -1 by taking exponential to the to the right hand side. We get why exponential minus two by two into t square minus which is the required solution of the aggression. All these clears your doubt and thank you.


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