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Exercise Solve the fallawing ncnhomogenecus differential equations using the method af Undetenined Coefficients 1. / Iv = 4r?2 /-3v' = Iv 2sin f3. /" - %v...

Question

Exercise Solve the fallawing ncnhomogenecus differential equations using the method af Undetenined Coefficients 1. / Iv = 4r?2 /-3v' = Iv 2sin f3. /" - %v' _ Av=el

Exercise Solve the fallawing ncnhomogenecus differential equations using the method af Undetenined Coefficients 1. / Iv = 4r? 2 /-3v' = Iv 2sin f 3. /" - %v' _ Av=el



Answers

Solve the differential equation.

$ 3 \dfrac{d^2V}{dt^2} + 4 \dfrac{dV}{dt} + 3V = 0 $

All right. So this problem again we're going to integrate the left hand side and the right hand side with respect to X. So I have prime prime X. D. X two out of by X squared dx the left hand side. We're going to get F prime of X. And on the right hand side where you can rewrite this to be two to the two times actually negative too. So we're going to add one to the power and we'll have negative two divided by X plus C. Yeah. And now we can plug in our initial conditions. So we have four Equals 2 -2 Plus C. And then we have c equals to six. So with this we get the F prime of X equals two negative two divided by X plus six. And we can integrate one more time just so we can get F of X. The X. And then well integrated. So on the left hand side we have F of X. On the right hand side we have negative to natural log the absolute value of X plus six X plus C. Are going to use case because we used the last time lets him find out a little bit before we go ahead and solve for K. So have F of X equals to the natural log of X. To the negative to power Plus six x plus K. All right now let's go ahead and plug in our initial conditions. Which was f. A one equals 23 So if the natural log of one is in fact zero. So we have three equals to zero plus six times one which is six plus K. And with that we get K two equal negative three. Uh huh. So by that we have F of X equals the natural log the absolute value of X. The negative to power plus six X minus three. They will be your final answer.

Let's solve this problem by solving for the homogeneous solution. So substitute out the Y double prime, the Y. Prime. And the white terms up with our square plus four R plus three equals to zero. We can factors out so we'll get that are plus one Times are plus three equals to zero. And this gives us our values negative three A negative one. So with that we can actually build our homogeneous solution is homogeneous solutions can be C one E. To the negative X. Policy to eat in a. of three x. And with that we can actually take a guess at our particular solution. So our guests for the particular assertion Y F P is going to be in the form A. E To the -3 x. Now note that there is a term here in the homogeneous solution that matches up with something in a particular solution. And since we don't want that, what we end up doing is we all end up multiplying by factor backs for our guests with a particular solution. And so now with that we can actually take the derivative of this twice, you know Y. P. Prime. It's gonna be E. So negative three X. With the product rule minus three A. X. E. 2.5 3 X. And Y. P. Double prime. It's gonna be negative three A. E. To the negative three X minus three A. E. To the negative three X. Plus nine A. X. E. to the -3 x. And so now let's simplify this this Y. P. Double prime term. And so this can condense into negative six A. Eat and I have three x. Plus nine A. X. Eat the May have three x. And so you can plug this into our original equation. And so original equation was Y double prime. So negative six A beats the knife three X. Plus nine A. Ex eats an F. Three X. Plus four Y. Prime. So plus four times a feat than they have three X minus 12 A. X. E. To the negative three X. Plus three Y. So plus three a. It's an area of three x. x. And this equals to eat the neck of three X. And so let's try and simplify this down before we build the systems of equations here. So are like terms you're going to eat this, this and this and the other. Like terms are gonna be this this oh sorry um And this place the X. In the wrong position there. And so now we can connect these terms or simplify them down. So have negative to a Eat. And they have three x. Plus well this plus this will cancel it against this. So this will be They have to a seat and a three x equals two Eating i. e. to the -3 X. Now we can divide by each of the 93x. On both sides. So cancel out. So I have that too. Two equals one Or that a equals negative 1/2. And so with this we can actually build our particular solution. Our sorry our total solution. So our total solution is a summer for homogeneous solution and our particular solution. And so are homogeneous solution in this case was C one Ethan negative X Plus c. two e. 2. And they have three x. And we just got our particular decision which is negative one half X. E. To the negative three X. And so this is actually her answer.

Okay, so if we let y b f f x why prime the f prime of X and y double prime the F double prime of X So given y double prime is equal to X to the negative three halfs From here we can solve why prime by taking the integral of X to the negative three halfs the X So here, if we find the anti derivative of X to the negative three halfs, that is why prime is equal to acts to the negative one half uber negative one half plus some arbitrary constants. See, now if we can just simplify, this will have why prime is equal to negative two x to the negative one half plus some arbitrary constancy. So I got negative to buy instead of dividing by negative one half a can multiply by the reciprocal which is the same as multiplying by negative two. So then we can use the constraint or the point that we were given f prime or y prime of four is equal to two. So first, if we rewrite this derivative, the first derivative is equal to negative two over the square of X plus arbitrary constants C So if we go ahead and try to salt sea here, so we'll plug in to and for the for the derivative and then plug in four and four X will have two is equal to negative two over the square root of four plus c. So with that we'll have negative to over two, which is negative. One ad negative. Want to both sides and look at that. C is equal to three. So why prime? Why Prime is equal to negative two over the square root of X Plus three Or we could rewrite it as negative two x to the negative one half plus three. Then, from here we can solve solution. Why so why is equal to the integral of the first derivative, which is negative two x two Negative one half plus three times DX So if we just find the anti derivative here, we'll see that we have y is equal to negative two x to the positive one half over positive one half plus three x plus some arbitrary constants. C So here, if we simplify this a little bit to make it look nicer, we'll have y is equal to negative for X to the one half plus three x plus some arbitrary constants. See, now if we use our other point that we were given were given the f of zero is able to zero. So if we use this, we can plug in zero in for X and zero in for y and soft seat. So I have zero is equal to negative for time, zero to the one half, plus three times zero plus c. So here we'll see that sees equal to zero. So hour y solution is going to be why is equal to negative four x to the one half plus three x.

Yeah, you can solve this differential equation by doing the integration. So the prime X is in the the integration from zero X die next. Plus, you have the power to x. It is is DX. Yeah, And once we get this equation as well, Mexico sign. Thanks. Plus one of the two needed. About two eggs plus C one. Right. So we get a crime zero, which is actually formed us one of the two C one. This is probably one of the two. Well, I guess you want this one. So affects we do this integration once again, get negative. Cosine X plus one over to eat about two x plus four to the immigration. So here we go for yeah, and we get negative sine X. That's one of the four need to powder two x x proceeded. Okay. Also solved this constant C two by using the fact that zeros So first we get This is zero. This is one of four the zero c two and this is equals one of four we get C two is zero. So here is zero. And this is how the solution looks like mhm


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