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Point) Consider the system of differential equationsdx dtdy =_5c_ dt~5yConvert this system t0 second order differential equation in y by differentiating the second ...

Question

Point) Consider the system of differential equationsdx dtdy =_5c_ dt~5yConvert this system t0 second order differential equation in y by differentiating the second equation with respect to t and substituting for T from the first equation. Solve the equation you obtained for y as a function of t; hence find as a function of t. If we also require I(0) = 2and y(0) = 1, whal are . and 9?x(t) y(t)

point) Consider the system of differential equations dx dt dy =_5c_ dt ~5y Convert this system t0 second order differential equation in y by differentiating the second equation with respect to t and substituting for T from the first equation. Solve the equation you obtained for y as a function of t; hence find as a function of t. If we also require I(0) = 2and y(0) = 1, whal are . and 9? x(t) y(t)



Answers

Find the general solution of the differential equation and check the result by differentiation. $$\frac{d y}{d t}=9 t^{2}$$

In the problem We have nine. Why nevertheless Plus four Y. is equal to booty. So Y. Of zero equals zero and wider. Zero equals zero. Now nine into esquire black clouds of white minus is Y. 0-. Why does zero plus forward lap less of one is equal to a bond is square. Now here we have nine S squared plus four. Laplace of Y equals two upon esquire so laplace of Y. Is equal to two upon esquire Hindu nine esquire plus four. So we have this as two upon. It's a square into nine into It's a squared plus 4.9 and further rehab. Y. F. They equal lap splice inverse of one upon to S squared minus three upon four into one upon. Yes it's required Plus two upon 3. What is what? So this is written as half D -3. r. sign to T upon. Mhm. This is yeah. D. Yeah so we have Y. F. D. As half The -3 before sign To 2.3. So this is our answer.

A matter Caribbean Our problem above fight one minus the square in the wild Apple dies minus to be violation plus two white equals zero lay off deep was the so via dies Beat equals one right double edged p equals zero So one minus T square in 20 minus two deep into one Does who minus duty plus two which is zero therefore one minus square into by a double life minus to be via dies Yes, toe weight equals zero If by off p equals therefore by off the equals is a solution. Thank you.

In order to solve this equation, we must first get into the standard form. Why? Prime plus p of x times. Why is Q backs and orgy this? We must divide by t squared to get this into the standard form that I just mentioned. So as I said, I'm dividing by t squared. Okay, Now that we've got this, we know we must determine the integrating factor. Each of the integral of three over tee times D t is the same thing has eaten the three, not a log of tea which is the same thing as eat the natural of t cubed. Now we know eats the natural log is once this the same thing as t cubed. We have just determined are integrating factor. Okay, Now we must multiply both sides. In other words, all terms by the integrating factor that we just determined in order to know integrate both sides and sulfur Why we have t cubed. Why is the integral of tea times one plus she squared d t which gives us t cubed. Why is one third times one plus t squared 23 over to pussy We use the power room or to do this other words. We increased our expert by one and we then divided by the new exponents. Lastly, we only want this in terms of why not any sort of co fishing in front? That isn't one, which means we must divide by t cubed for the terms in order to derive our final solution plus c times t to the negative three which the same thing is pussy over too cute.

Mhm. We want to solve given differential equation which states 90 x minus x squared dy is equal to 90 Y. This question is challenging our ability to solve differential equations in particular. It's challenging our ability to use integration techniques in conjunction with our newfound technique for solving separation of variables. So separation of variables requires three steps to execute and perhaps one will isolate X and Y terms on either side of the equation. So why terms were going less actions on the right and step two, we're going to integrate both side of the equation which is valid because each side will have a differential dy dx and step three we're going to solve evaluating each integral with integration methods. So step one, isolating gives do Y equals D X over one plus 19 X squared. Next integrating both sides gives integral dy equals integral dx over one plus 19 X squared. The right hand side is an arc tangent. And so we evaluate this integral is as follows, why is equal to the arc tangent of one third X plus C. The constant of integration.


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