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QuestionFind the function y y(c) (for € > 0) which satisfies the separable differential equationdy dc+172 I > 0 ry?with the initial condition y(1) = 3.A...

Question

QuestionFind the function y y(c) (for € > 0) which satisfies the separable differential equationdy dc+172 I > 0 ry?with the initial condition y(1) = 3.Add WorkCheck Answer

Question Find the function y y(c) (for € > 0) which satisfies the separable differential equation dy dc +172 I > 0 ry? with the initial condition y(1) = 3. Add Work Check Answer



Answers

Find the general solution of the differential equation. $$y^{\prime \prime}+y^{\prime}+3 y=0$$

Problem. Seven of section seven Point want us to the solution of the following differential equations. So the first thing that we're going to do is we're going to rewrite this week y prime as D y DX. We're then going to multiply DX to both sides so that we have the very rules isolated. So that leaves us with the Y is equal to negative three DX. Now we're going to integrate both sides. That leaves us with Why is equal to negative three x plus c. The last thing that we have to do is we have to come back to this initial condition and satisfy it. So to do so, we need to set X equal to zero and why equals three. So writing that out we get three is equal to negative three times zero plus e. So this means that C is equal to three. Our solution is why is equal to negative three x plus three

In this problem, we need to solve the differential equation. De vie by dx is equal to X divided by Y. With the initial condition that why not? So y of zero equals negative three. So first we can separate the variables and that will give us why dy equals X T X. And from here we'll just integrate on both sides. And that will give us from thee baba rule for anti derivatives, Y squared divided by two is equal to x squared divided by two. And then we have to pat a constant of integration will call it C. And we can multiply the entire equation by two. I said let's call it C one and we get y squared equals x squared Plus two Dimes C one. And since the constant of integration is arbitrary, you can write this as y equals x squared plus. See now we can take the screws on both sides and we have to take the plus minus square root because we're introducing a square root and were given that the initial condition is actually negative. So we'll consider only the negative square root. And now we'll just substitute in our initial condition which is why not equals negative three. So why is negative three? When X is zero mm. That gives us negative square root of C. Being equal to negative three. And we can square both sides. And that gives us nine equal. See which means that our differential equation solution is negative X squared last night

Okay start with this problem I'm gonna go ahead and move to E. To the X. The left side. And I'm gonna rewrite why Prime S. D. Y. T. X. And I'm gonna move dx the other side as well. So why d why? Of course to either the X. Dx and now I can integrate both sides. So this is gonna be y squared divide by two. And on the right hand side is just gonna be to E. To the X. Policy. Now I can multiply by two. So I have four E. To the X. Plus C. And I'm going to go ahead and plug in my initial condition here which was why A zero equals three. It's gonna be three squared Equals to 480 power. Just one part. It's four times 1. So it's just before plus c. And so Have nine equals to four plus c. Subtract four C equals 25 So we can plug that in. So we have y squared equals to four E. To the X plus five. And if we wanted to simplify down a little bit further, we can take the square root, so have Y equals to the square root four E. The X plus five, and that'll be your answer.

Problem. Two of section 7.1 asked versus all the following differential equation. So the first thing that we need to do is we need to separate the variables in this equation appear so to do so we're going to write Why Prime Misty y DX Brennan said that equal to three wife, we're then going to multiply both sides by D Y and divide those sides by. Why Sorry. We're gonna multiply both sides by D X and multiple and divide both sides by Why? So that will leave us with D Y over. Why is equal to three d x? So the next thing that we're going to dio is we're going to integrate both sides. So when we do this, we will be left with Ellen of the absolute value of why and that will be equal to three x plus c. So now we need to get rid of the Ln of the why. And so we get why by ourselves and to do this, we need Teoh have a base of E. So when this is all worked out, we that leaves us with why is equal to eat to the three x plus sorry. Times E to the C. Now we're just going to replace this with a variable A, um and that will leave us with the equation. Y is equal toe e to the three x times A. So now we need to satisfy this initial condition where in why of zero is equal to negative two. So we're going to set x zero and why it's negative too. And that'll give us eat to the three time zero times a is equal to negative two and eat to the zero. Anything to the zeroth. Power is equal to one. So we get one times a which means that a is equal to negative two. And then our equation will look something like this. Why is equal to negative, too? Times he'd to the three x.


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