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Solve the intial value problem.
$(1-t) \frac{d y}{d t}-y=0, \quad y(2)=-4$

Of this initial value problem. Let's integrate. So we have s equals and Billy could rewrite the STT as three t squared minus to t. So it's easier to integrate. We get t cubed minus t squared, plus c a constant. And we have to find sea by plugging in r point as equal zero teeples one if you plug that in zero equals one cubed is one minus one squared is one class c and we get that C just equal zero so we can rewrite as us s equals key cubed minus T squared plus C again or plus zero So we don't have to do anything and then this is our solution.

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In this video, we're gonna go through the answer. It's a question of a 23 from chapter 7.5 to us to find the solution in the class space off this initially problem so that in her Virginia's on the right inside this function G is a piece wise, continuous function, meaning that the transform least exists that show. Um, let's just first workout. What little class transform off off dysfunction? G is okay, so it's the integral from north to infinity off e minus s t times J did see. Hi, guys. This is gonna be split up into two into Goes not to to t minus. Testy Did see plus Thio Infinity 05 He's the man Necessity, Petey His first term at so it's first integral. We can use intubation my parts So it's going t times One of the minus s eats the minus s too. So with left the tears that isn't on with integrated the e to the minus zesty that's evaluated it zero to minus the integral off differentiate t. This is just one ah times by the integral of beat the fantastic. That's times by minus one of s the minus the minus becomes a plus. Introducing 02 off E to the minus two. It's the minus s. We could take a five. Oh, this, uh, right hand it to go on, Dhe. Let's see. We're gonna have one over minus s. Okay, a to the minus. S t between team an affinity. Okay, this first term, when it's evaluated, T is equal to zero. Then that time's gonna be zero. So we just need to do that too. Minus two over S E to the minus s. Who's the minors to this, uh, this next into grow. It's one of s outside, minus one of S E to the minus esty between zero and two. When s what is equal to infinity? This right hand term zero. So we only need to do. Have I wasted too? That's where the fire of S E to the minus s t needs a minus. Started. Take Bess, but Ty's equals two. Okay, So this guy, this guy could be combined to be three of S e to the minus. Tootie on DDE there's a tear in the middle is equal to minus Gonna best squid sense e matters to us. plus one. Okay, so now we have that as I were, we must transform of the right inside. So we take those responsible off the whole equation. Whole different equation on using standard questions for rules. That's s squared times. Why, big wise, the Rebus transform off. That's why. Then, minus minus warms. That's plus one times desk. Then the initial. Why prime is zero that cost four times why that's equal to this three of us eats the minus 22 minus one of s squared a monster. T this one. No, I think I've got my essence of my teeth mixed up here one second. Okay, so this team should be on Bess. Maybe because you're now in the bus space. Yeah. This tea on this too. Uh huh. Should be SS. Okay, that could be arranged for counsel. Why? So we're gonna have three of s East Minister to s minus one of us squared. Ease the minds to us. Maybe that's what Because attract us about sides. Divide by the coefficient. Why? Which they're squared. Plus four. Okay, just write this out slightly more neatly. Bust three times s times E to the minus two s So we're doing here is most blank denominator numerator by a squared as a minus E the miners to U s minus one minus s to the par three old wide by s squared Times Square. Possible, huh? On that is our solution into space.