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4) Solve the following differential equation (I+ U)dr - (I - v)dy = 0 _dr tan -'(r) 1+0You mny nccd thc iutogral...

Question

4) Solve the following differential equation (I+ U)dr - (I - v)dy = 0 _dr tan -'(r) 1+0You mny nccd thc iutogral

4) Solve the following differential equation (I+ U)dr - (I - v)dy = 0 _ dr tan -'(r) 1+0 You mny nccd thc iutogral



Answers

Solve each differential equation. $$y^{(4)}-y=0$$

Right. So sir, this problem by rewriting the Y. Prime in the form of dy DT that equals tante. And now what we can end up doing is we can split this dy DT by multiplying by factor of DT on both sides. So all that look like is Y squared dy equals to 10 T. T. T. And to get rid of these DTs. And the wise we're gonna integrate. So let's integrate both sides. And the integration of why squared of course is why cubed over three. And the integration of tante is the natural log of second. Mhm. And don't forget Plus C. And now what we can do is we can multiply by a factor of three. So I have three natural log second T. For C. And we can take the cuba so cube root, three. Natural log seeking T. Policy. And we can rewrite this in the form of Y equals cuba three natural log of one over co sign T plus E. And one of her co sign of T. Is equivalent to co sign of T. To the negative one power. And so when we have a natural log of this natural log of coastline of T. To the negative ones power, we can do is we can bring this negative one to the outside and so we'll have negative natural log of course 90. And so this final answer will look like yq. Sorry Y equals the key route of negative three natural log of coastline. T. The absolute value of coastline T. Plus C.

Good one. Today we're going to solve problem number 19. The given Pasztor equation is violence plus right planets because seek X plus call. Six. So this is be or fix and this is Kyul fix. So whilst zero equals one, so integrating factor equals a integral panix D F, which is a quarto. See kicks so white in Do Seek X equals Integral CKX. Seek X plus call six DX for right equals Call six in the girl six square it's plus one the X So bye equals call six in do Panix plus X plus c So y equals so next plus Xcor. Six. Let's see because X this is the general solution. So boundary condition is given like y zero equals What? So one equals zero plus zero plus e Very good C equals one. So why equals? Sorry. Next plus X call sex plus call six. This is the specific solution. Thank you

All right. So this time we are working with the differential equation of the second, distributive of why -4 times the first derivative plus four times Y. B zero. And so this one is a little bit unique because when we factor it for working with the auxiliary equation, which is going to be r squared minus four R plus four. Factoring this gives us ar minus two square, right? Because um negative two times negative two of those A positive four. And then anyway, so we have is that our our one is equal to R. R two which is equal to two. So when we go to put it in our general solution, we're gonna do something a little bit different uh in the past few problems. So we'll take Y and we're going to take the first step is the same and we have a constant times E. To the power of their first are two times X. And we add that to a constant. But this time due to the multiplicity, we're going to take that constant times X. And multiply that by E to the power of two X. Okay, so the general solution has this extra X in here. So it's good to be aware of this property when solving these differential equations. Otherwise, this is going to be our journal solution, since we have no initial values to solve it any further.

Talk about question about 63 way we need to solve this differential equation. So, uh, looks divide both sides by a four plus 10 square X So we have sex square X over four plus pan square X Why does has nothing but D y or DX so we in fact, we integrated both sides. So this can get it on us. Uh, why dash, we integrate with respect to work. So we have wired ash integrated with respect works. There's nothing but sex were x b x or mhm four plus dance square x So integration, all wired dash will just be wired Onda here we made a substitution off can access t because if we differentiate, we have sex square x d excess d t So the integral in terms off comes out of the sex were X dx and really pleased by DT So we have DT over four plus the square. So this comes orders. Why is equal to d t Over are two square plus t square. So DT over two square the city square is nothing but a straightforward formula off can in words. So this will be 1/2 Dan and worse the or two plus see where t is nothing but panic. So the final answer comes out as why is equal to one or two Annan Waas annex or two because the concern of ridiculous see, So this is a final answer.


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