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Let p is the smallest nonzero_ and 4 is the largest nonzero digits of the last four digits of your student ID numbcr;Qucstion L Solve the following ordinary differe...

Question

Let p is the smallest nonzero_ and 4 is the largest nonzero digits of the last four digits of your student ID numbcr;Qucstion L Solve the following ordinary differential equations A) prsec B) =Sy(r-2)y

Let p is the smallest nonzero_ and 4 is the largest nonzero digits of the last four digits of your student ID numbcr; Qucstion L Solve the following ordinary differential equations A) prsec B) =Sy(r-2)y



Answers

Let $N=2+22+222+\ldots$ upto 30000 terms. (a) last four digits of $\mathrm{N}$ are 8580 (b) last three digits of $N$ are 580 (c) last four digits of $\mathrm{N}$ are 9580 (d) last digit of $N$ is 0 .

Mhm problem. We wish to solve the differential equation DP DT equals p minus a. This question is challenging and understanding of differential equations in particular how to find solutions to such equations via the separation of variables methods this method for solving different use it requires to proceed in three steps and step one, we isolate the X and Y. Or in this case P and T terms on either side of our equal sign. That's the separation of variables. Instead to will integrate both sides of the equation and in step three will simplify the equation as necessary. He does have one. We can read our folks uh D P overpay right as equals D. T now P and left hand side. And he is on the right. That's because both sides of differential we can integrate as the integral. Dp over p minus eight equals integral. Et This simplifies this algorithm. P minus eight equals T plus constant C C. Is a constant immigration. Remember thus, if we exponentially both sides to get rid of the algorithm, we have PMS A equals B. To the E R C. I have written into the C. S. A. Thus adding a double size gives our final solution. P is equal to capital A. Even the T plus A.

I in this question, we are asked to fight the last digit off. I s cold. Uh, for these four chords and the isis has the formula like this one, it will times it's did it with the number off their position plus two. So the first stage is Tom Tree and the last digit times nine. And we have to form like us so we can just calculate them. I do show the 1st 1 and you can fi everything else using this formula as well. So we put this in Ah, yeah. There is no other way but to calculate this. So tree four times five is 20. So is nigh. My 11 is 35. So is too six disappear seven Ed is 56. So this one for the ed. So it's four and 72 is sinks, right? Yes. And so it will become tree Indian. You smart 11. It would become tree. So the last digit off the first court would be tree. And similarly, you can do this for every other cases you will have. And I said like this. Oh, so this is a B C d. Isn't it sorry on the a boo. The A will give tree as we calculate it be Will give 10 wish in this case they awry ass ex So is 10 months 11 But we want to put it in one digit so we can put one and zero to get there. We will use the letter X like in Roman number. See you give Ni and you give tree. If the calculation is like If this doesn't match with yours then something goes wrong here. But it should be really symbol We miscalculate sometime. Don't worry. That is the answer. Thank you.

Problem we would like to solve a given differential equation. DP DT by this. A P equals B where A and B are concerts. This question is child, you understand the differential equations in particular? How to solve them? Be the separation of variables method. We proceed in three steps to execute this method properly and step one we isolate are X and Y are in this case are P and T terms and either side of the equal sign. That's our variable separation. Instead, do we integrate both sides? And in step three we simplify our equation unnecessary. South procedures have one DP DT would be plus ap we isolate now piano left on the right as DP over people to be over A equals 80 T. Next proceed to step two. We can integrate both sides. The integration gives L N P plus B over A equals 80 plus C. Thus exponentially both sides, you have people over equals 84 C. We can rewrite either the C s A or capital A rather that's A E a T. Thus we can now subtract B over A from both sides give our final solution, Which is P equals capital au to the 18 -7 over a.

Were given a question about three digit numbers and were asked to answer this question using counting principles. Now we're giving conditions and were asked and under each condition to answer how many three digit numbers that we conform. So in part A, the condition is the leading digit cannot be zero. So we have a number with three digits. I'll call them A B and C, and we know that a cannot be zero. Therefore, by the fundamental counting principle, the number of ways we conform a three digit number under this condition is a number of ways to choose a well, since they cannot be zero. There are nine choices for a He could be anywhere from 0 to 9. So be it could be any of 10 choices times number of ways to choose C, which is also 10. So this is 900 numbers now in part B. The condition is that the leading digit cannot be zero, and no repetition of digits is allowed. Once again, we have a three digit number, ABC and again A is non zero, and the condition that no repetition of digits is allowed means that they cannot be equal to be and B cannot be equal to see and they cannot be equal to see. So by the fundamental counting principle, the number of three digit numbers we conform it for this condition is the number of choices for digit A. Well, if you start with a, there are choices one through nine, there's nine total choices. Then for digit B, we have all the choices from one through nine, except for a sorry from zero through nine, except for a so beacon be any of nine different options. And finally, the number of choices for see well, we have our 10 choices. But then we know that C cannot be equal to a So now we have nine choices left, and we also know that C cannot be equal to be so. We have a choice is left. So times eight So we have nine times, nine times eight or 648 numbers. We conform under this condition in part C. The condition is that the leading digit cannot be zero, and the number must be a multiple of five. So again we have a three digit number. ABC A cannot be equal to zero And if the number must be a multiple of five, this means that the last digit see must be equal to zero or must be equal to five again. By the fundamental counting principle, the number of three digit numbers we conform to this condition is the number of choices for a, which again is nine times the number of choices for B, which we see ca NBI any choice from 0 to 9 or 10 times the number of choices for C, which we see there are only two options for see either zero or five, so times too. So this is 180 three digit numbers. Finally, in part D. The condition is that the number is at least 400. So we have a three digit number. ABC and ABC is at least 400 now. What does this mean in terms of the digits? But we have that a must be greater than or equal to four, and that's really the only restriction we have. Okay, So again, by the fundamental accounting principle, the number of three digit numbers we conform under this condition is the number of choices for A which appear to be four through nine. So there are six choices for a times the number of choices for B, which is zero through nine. So there's 10 choices for B times the number of choices for C, which is also 10. So this is 600 so there are 600 three digit numbers getting this condition.


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