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Find yy as a function of tt if 3y″+28y=0,3y″+28y=0,y(0)=7,y′(0)=5.y(0)=7,y′(0)=5. y(t)=y(t)=...

Question

Find yy as a function of tt if 3y″+28y=0,3y″+28y=0,y(0)=7,y′(0)=5.y(0)=7,y′(0)=5. y(t)=y(t)=

Find yy as a function of tt if 3y″+28y=0,3y″+28y=0,y(0)=7,y′(0)=5.y(0)=7,y′(0)=5. y(t)=y(t)=



Answers

Show that the function $f(t)=\left(e^{-t}+1\right)^{-1}$ satisfies $y^{\prime}+y^{2}=y$ $y(0)=\frac{1}{2}$.

This is a question of 22. If are off doing is a good war of day to symbolize the one is equal to D to Then you consider that the function is 1 to 1 right now substitute divan, four feet. So here are over 51 is equality D windows to four minus one and here are lofty to physically do. Do you do this to for minus one now convertible dysfunctions. So here we have our of the one is equal to our off to right now You're our lofty one. That is accord with you in just two for minus one. And here are off the truth. That is equal toe. Do this two for minus spot. Right now you're negative One and negative one will begin. Set out. So here we have a believe he wonders to four is ableto Do you do this to for right now? Take four through both sides. So you are The one is equal to plus or minus. Do you do against as that you cannot be able to? He drew for some values off d night. So we consider that the function is not run to one ocean, right? Thank you

In order to find the function are we're going to need to find the integral of each component. So let's do the first component. We're taking the integral of eat of the teeth. Okay, VT. So then the integral of e to the T is gonna b e to the T tips, and then we're gonna have a plus C one. And remember, at zero, this should be sorry, an r of zero. Now, when t is equal to zero Ah, the first component to go to two. So we're gonna have eat of a zero plus C one is equal to two. And remember, eat of zero is equal to one. So a move that over you get that C one is going to be equal to one. So are our t is going to start with. First, we have e to the T plus one. Okay. And then the next component we're going to do with the integral of lips integral of scientist E. Sorry about that. Okay. Sign of t d. T Integral of sine is negative. Come sign of tea and then we need to add a plus C two. So again, at zero, we're having to hear. So we have that negative negative. Uh, co sign of zero plus C two is equal to two. Now, remember, co sign of zero. This is equal to one, uh, cause I Nazir's able to once we have negative one plus C two equals two. So then when we add this over to the other side, we get that C two is equal to three. So our second component is going to be equal to negative, cause I in t plus three or I'm gonna write it just as three minus cosigned of t. And then finally, for the last component integral of seeking squared T DT. Well, since the derivative of tangent is seeking squared, then the integral of seeking squared is going to be tension. We're gonna have tension of tea then plus C plus a C three and again, remember, uh, we're having tension of zero. Plus C three is equal to two. Attention of zero is zero. So si three equals two. So our last part is gonna be tangent of T. Uh and I'll do it as our right. It has actually two plus tangent of T. So this is going to be our our of tea here

In this problem we have D Q by DT is equal to K. Q. Now the value of who does he can be written it. Hey you the, the function need Hubie is equal to three to the power Katie. We'll see the constant. Now let us check it so I can write develop you dusty two dash T is equal to C. E. To the power Katie multiplication K. Which is equal to gave you the value of two deaths. He can be written edge, take you the so solution is the solution. But this problem is U. T. is equal to see to the park 80 as our final answer.

So for this question were asked to determine if y is a function of X. The first thing that we want to do is go ahead and isolate X. So in this case we're going to have to square root both sides. And remember when we square root, we also always have to consider the positive and negative solutions right? Because if we have four equals X squared tend to play X can be positive or negative too. So we have something like this. Right? And so in this case we know that if y is a function of X, any input value has to be matched with exactly one output value, right? If it's matched with two or more output values, liberation is not going to be a function. So if we assume you know X to be, let's say three, we know that. Why would equal positive or negative nine minus one? Which is eight. So in this case we have one input value of three but we have to output values right? It can be positive or negative square root of eight. So because of that, we know that it's not going to be a function. It can only that one input can only be matched with one output. And that's not true in this case, so it's not a function.


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