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Fl Find particular solution of 2y +y 2 sin 3x + cos...

Question

Fl Find particular solution of 2y +y 2 sin 3x + cos

Fl Find particular solution of 2y +y 2 sin 3x + cos



Answers

Find $D_{x} y$. $$y=2 \sin x+3 \cos x$$

So for this problem, we are going to start with our equation wise, equal to a co signed two teeth, plus be signed to T. And because we know that we are finding a particular solution to our differential equation, we're going to need to find the first derivative of this function. And when we do, we get a Y. Prime is equal to negative too, eh? Signed. Two teeth plus two b co sign two teeth. Um, and I just did this using my regular rules of differentiation. Um, so we had to do can rule and each step here. So the derivative of coastline is negative. Sign times. The derivative of two tea will be times two. So that's how I got each of my two functions there. And, um, now we want to substitute this into our differential equation to find our particular solution. So our differential equation is, um why prime? Plus why the left side? So that's negative too, eh? Signed Two teeth plus two b Co sign to t from r Y Prime I'm plus why? Which will end up being plus a co sign two teeth plus be signed two teeth and this is equal to on the right side of our equation. Four Sign, two teeth and so to Seoul for our A and B values. I'm essentially going to combine my sign terms and co sign terms. So my sign terms I can go ahead in factory. A sign of to t This will be times negative too, eh? Plus B. And for my co sign terms, um, we can again factor out a coastline of two teeth so we'll write, Plus coastline of two teeth times. This time it'll be times to be plus a, and this is still going to be equal to four. Sign to tea on the right side of the equation. And so, from here, um, it might look a little bit complicated to figure out what our A and B values are going to be. But if we write this as plus zero co signed to tea on the right side, we can see that essentially, um, we have a system of equations that we're going to have to solve, so we know that a negative to a plus B is going to have to end up being equal to four for this coefficient on the right side and to be plus a will have to be equal to zero again for the coefficient of the coastline to t term on the right side. So when we set this up, we said negative too, eh? Plus, B has to be equal to four, and we have the second equation arm to B plus, eh has to be equal to zero. So I'm going to, uh, solve for a in this equation on the right. Here we get a is equal to negative to be. So when I substitute that into our equation on the left over here we get negative, too. Times negative to be, um plus B is equal to four. And so a negative two time negative too, will be four Beat plus B is equal to four, which will simplify to five bees equal to four. And finally B is equal to 4/5. And now I'm just going to substitute that into my equation. On the right side, we get a is equal to negative eight over five here, so I'll go ahead and make a new page for our final solution. All we have to do is plug our A and B values into the Y function that we were given. So we get, um our particular solution is wise equal to eight, which we said was negative eight over five times Co sign of two teeth plus B but we said was 4/5 times the sign of duty.

The no D. Y. Writing the answer why equals cynics plus Cosine X cubed. Dy is going to be the derivative of that D. X. Derivative of that is three sign X plus Cosine X. To the second power times the derivative of the inside, which is going to be driven to the sinus Co sign derivative coastline is opposite of sign. And that's it. I should probably erase here and here because it's not necessary. But that's it.

Hello welcome to the snack is in this lesson. We would so the salt for the general solution of the differential question by using the into Britain documented would write a the way that can be compared to the general fund and that is by dividing both sides. Bye. Thanks. So that the coefficient of why prime would be one. We can compare it to the general form. Yeah. Okay at this point we compare it with. Mhm So he F. P of X. This according to our necks. Okay. Mhm And cube X is equal to cost X. X squared help were introduced an integrate the basis for integration factor. That is the integral. He affects B. X. Meeting grow two on X. B. X. And very sequel to mhm Okay two Lennox or learn ex squint. Yeah. Okay so with this we would have Y which is equal to one on eat the prop. It affects the integral E to the X. Kale of X. Dundee X. Mhm mhm Yeah but oh okay. Okay. Okay so that is Why which is go to one own mm to the power Lenox square mentee girl eat so let X squared the whole of the kill X. Is course thanks on X squared. What is your ex? So why she calls you one? It is a X good land X squared Okay course X. Call the X square new X. Now this across all that so that we have Weitz got one on expert and to grow cost X. Dx Why I support you one. So mm that is sign X last constant. So you have why? As you called X minus Actually five negative two sine X blast. Excellent. Five negative two. I'm seen or would bring the C forward. Oh, yeah. Yeah. So thanks for the time. This is the end of the lesson. Mm Yeah.


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