Let y(x) be the position of a particle at time x. Suppose that we know that the
ID: 2846213 • Letter: L
Question
Let y(x) be the position of a particle at time x. Suppose that we know that the velocity of a particle satisfies the differential equation y'(t)=ty(t) with y(0) = 1. We will try to give a reasonable method to approximate some of the positions of the particle. 1. Integrate both sides to show that y(x)= integral from x to 0 ty(t) dt +1 2. Since y(t) is not known we cannot compute the right hand side. Instead, we can try to offer approximations similar to the techniques used in the text. First, let y0=y(0) for notation use. Let us approximate a value for y(1): y(1)= integral from 0 to 1 ty(t) dt+1 approximately =integral from 0 to 1 ty_0 dt +1 approximately= 1/2 +1= 3/2 Let y1 be the approximation to y(1) computed above. EXPLAIN why we might expect this to be a reasonable approximation. 3.We can use it to help us compute an approximation to y(2): y(2)= integral from 0 to 2 ty(t) dt +1 = integral from 0 to 1 ty(t) dt+ integral from 1 to 2 ty(t)dt +1app approximately = integral 0 to 1 ty_0 dt + integral 1 to 2 ty_1 dt +1 FINISH the above computation to give an approximation to y(2). 4. Compute an approximation to y(3) 5. How can this approximation method be improved at all? 6. Explain how a numerical technique could be set up to approximate solutions to the differential equation Integrate by parts.
This question was asked/answered already, but there was no explanation for parts 3 or 4. That is the only thing I don't understand. If someone could explain to me why it is acceptable to use y(0) as an approxamation for y(t), given the context of integral approximation techniques like the trapezoidal or midpoint rules, or Simpsons rule, that is what I don't understand. Thanks.
Explanation / Answer
Let y(x) be the position of a particle at time x. Suppose that we know that the velocity of a particle satisfies the differential equation y'(t)=ty(t) with y(0) = 1. We will try to give a reasonable method to approximate some of the positions of the particle.
1. Integrate both sides to show that y(x)= integral from x to 0 ty(t) dt +1
2. Since y(t) is not known we cannot compute the right hand side. Instead, we can try to offer approximations similar to the techniques used in the text. First, let y0=y(0) for notation use. Let us approximate a value for y(1): y(1)= integral from 0 to 1 ty(t) dt+1 approximately =integral from 0 to 1 ty_0 dt +1 approximately= 1/2 +1= 3/2 Let y1 be the approximation to y(1) computed above. EXPLAIN why we might expect this to be a reasonable approximation.
3.We can use it to help us compute an approximation to y(2): y(2)= integral from 0 to 2 ty(t) dt +1 = integral from 0 to 1 ty(t) dt+ integral from 1 to 2 ty(t)dt +1app approximately = integral 0 to 1 ty_0 dt + integral 1 to 2 ty_1 dt +1 FINISH the above computation to give an approximation to y(2).
4. Compute an approximation to y(3)
5. How can this approximation method be improved at all?
If we take smaller and smaller time intervals, founded numbers will be more precise.
6. Explain how a numerical technique could be set up to approximate solutions to the differential equation Integrate by parts.
As we saw above, we found y(t) for t=1 and t=2 and t=3; if we go ahead and find y(t) for every t (approximately), in fact we have been solved the differential equation.
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