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Q3. Define and solve the following DE using U.C methoddly dy +4 + Sy = x + e-2xcosx dxz dx...

Question

Q3. Define and solve the following DE using U.C methoddly dy +4 + Sy = x + e-2xcosx dxz dx

Q3. Define and solve the following DE using U.C method dly dy +4 + Sy = x + e-2xcosx dxz dx



Answers

Find dy/dx for the following functions. $y=\sin x+4 e^{0.5 x}$

Hello Friends. We have to solve the defensively question. No, By following the method of example two. Yeah. Okay so the defensive equation is given that This car by plus 40 by You close to zero. So we can decide the In two d plus four. Very close to zero. Mhm. So we will replace the bicycles to zero. So D plus forward. Do you buy calls to close to zero? So we can solve it. Do you set up on the X costume minus of what is that? So we can solve it. They set upon for that. It comes to matters of the X beta surplus and very well method we can solve it one up 14 one upon the integration of this will be l injured Because 2 -1 bless Ellen seven this is the constant. So we can I do when I go on for you can also write this like this sorry we can add minus of X Plus constant. So we will write the constant in the form of Ellen that is like this Ellen of seven. Okay so we can write Ellen off Jade upon even it was two miners of X. So zero upon seven will be minus of four X. Helen. So Ellen there will because to so that will be costume seven into the to the power of minus four X. And now we will replace the value of jail that is need to buy that is dividing Dx develop on the X. Is constant seven and to leave our minutes of four x. So diva equals to seven into departments forex into d. x. No again we will integrate this so this will be a costume My equal to seven Upon -420 Power -4 X plus you too. So why will be question this is a yes this will be the new question that is even this The department is four x plus. You do. Okay. So solution of this given defense electrician will be by calls to seven by question C. Two plus seven days into departments forex. So this is the answer I hope you understood. Thank you.

Hello. This is the defense of the question. We have to solve this. So we can adduce. Carmen is three D plus two Or y equals to zero. So we can die. They managed to multiplied by The man is one of my close to zero. So we will replace the management buy into vehicles to zero by that. Okay, so they managed to into Z equals to zero. So we can do about it. Do you set up on the eggs money? two said it goes to 0? Yeah. So this will because to desert upon that because to two D. X. So this will be caused to l injured money because of the works plus L. N. C. One. So this will be a challenge ahead upon Stephen it causes you to X. So jade upon seven will be into the power of two x. So jade will be into the power of two weeks in 27 So now we will again. All right. Mhm. In the place object we can die. Demon has run into by. So, the man is one into why He calls to seven into the power of two weeks. Yeah. So we cannot divide Man is what he calls us. Even into the power to X. This is the first order. This is a linear equation of the first order. So we can write the solution of this. Why into into the power P. D. X. He calls to you into into the power of pd X. Of the X. Bless you discussion PS -1 here. We can knighted dy minus of I. D. X. Sorry, this will be dave I minus Y. Just to see one into the power to X. Of the X. This is the first order linear differential equation. This is the Solution PS -1 here. So you two departments of one. The exact cause to Q is is even into the power two weeks into. Into the Power of man is one of the X plus. You do over into integration of management in the years. That is 16 interiors and no one has one. Believe it to the power of minus X. He calls to even into the power to work and to the power money sex do you express? So you too. So this will be a cost to buy into ET power Manager because to seven into into the power of Yes. Do you explore c. two further? We can solve it. So this will be cause to buy into it departments. Agriculturists even here to the public's plus you do so by will be Stephen into the power two weeks Plus You two into the power x. So this is the this is the answer of the linear differential equation. This is the solution of the linear differential equation. I hope you understood. Thank you.

So a section 4.5 problem to 41. We're dealing with a linear, ordinary first order differential equation, and we have a boundary condition. 50 is equal to three on. So what we need to do is just to put this in standard form. And we say, Well, it's already in standard forms. A standard form is why prime some function of X times. Why is equal to some other function of X so I could find the integrating factor is going to be e and so the coefficient of y just one. So this is going to be e to the X is the integrating factor. So that tells me I can rewrite this differential equation as why into the X prime is equal to X e to the X, Then the solution would be to integrate both sides of this equation. So in integrating both sides of this equation, I end up with why e to the X is equal to either the ex text minus one plus a constant of integration then to solve for why it's just a divide by either the x o why is equal to X minus one plus c e to the minus X. Then you have the boundary condition that why of zero 2nd 3 that tells me that to solve for C, I'm gonna have a three. It's a people to zero minus one. See? And then each of the zeros just simply one. So that tells me that C is equal to four. So the final answer here is why is equal to X minus? One was for E to the minus X, and that is the exact solution to this differential question.

Okay, so let's take the fall in derivatives to notice that our outermost function is this power of three. Still, we have bigger three and in our inside to the power of three minus one Said it to an end times the derivative of singer, which is kohsh of its inside, which is four x and in times a derivative of forex, which is for okay, so we have four times three. So that's negative. 12. And we have singe square of four x and kohsh of four X. Okay, so this is our solution.


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