# Exact solutions and the convergence of eulers method

• thedude36
In summary, the conversation discusses the question of finding the exact solution to a given initial value problem. The person is having trouble with part a and has been trying to use the integrating factor method. They present their attempt at a solution and ask for clarification. The expert confirms that their method is correct but points out a mistake in their calculation. The person thanks the expert and resolves the issue.
thedude36
im having trouble with this question - http://i.imgur.com/Ars4J1b.png - more specifically with part a, as i have a good idea how to go about b. given the initial value problem

y' = 1-t+y , y(t0)=y0

show that the exact solution is

y=$\phi$(t)=(y0-t0)et-t0+t​

we've only spoken of approximations in class and I've just been kind of guessing as to how i should go about it so far. I've tried to look up the term exact solutions but haven't found anything of much use.

## Homework Equations

integrating factor: if dy/dt+ay=g(t), then μ(t) is such that dμ(t)/dt=aμ(t). Multiply both sides of the equation dy/dt+ay=g(t) by μ(t) to obtain μ(t)dy/dt+ayμ(t)=μ(t)g(t).

## The Attempt at a Solution

rearranging the given formula, i was able to get

dy/dt-y=1-t​

where a = -1. thus, dμ(t)/dt=-μ(t) making μ(t)=e-t. multiplying bothsides give

e-tdy/dt-e-ty=e-t-te-t

the left side can be obtained by the power-rule if the initial function was d(ye-t)/dt so we replace the leftside with this

d(ye-t)/dt=e-t-te-t

integrating bothsides and then solving for y gives

y=-2-t+cet

however, i don't think this is the correct method. What should i be doing instead of this?

Your method is fine. You just have a mistake calculating your last line. Re-check that -2-t.

LCKurtz said:
Your method is fine. You just have a mistake calculating your last line. Re-check that -2-t.

Thanks! I am not sure how i was integrating that wrong, but a quick check on a calculator quickly resolved the issue. I appreciate it!

## 1. What are exact solutions in mathematics?

Exact solutions in mathematics refer to the solutions that satisfy the given conditions or equations without any approximation. These solutions are considered to be the most accurate and precise, as they are found through analytical methods or mathematical calculations.

## 2. What is Euler's method and how does it work?

Euler's method is a numerical method used to approximate the solutions of differential equations. It works by breaking down the continuous curve into small linear segments and using these segments to estimate the value of the function at a given point. The smaller the segments, the more accurate the approximation will be.

## 3. How do we determine the convergence of Euler's method?

The convergence of Euler's method can be determined by checking if the difference between the exact solution and the approximate solution decreases as the step size decreases. If the difference approaches zero, then the method is considered to be convergent.

## 4. What are the advantages of using exact solutions compared to Euler's method?

The main advantage of using exact solutions is that they are more accurate and precise compared to the approximated solutions obtained through Euler's method. Exact solutions also provide a deeper understanding of the problem and can be used to verify the results obtained through numerical methods.

## 5. Are there any limitations to using Euler's method?

Yes, there are limitations to using Euler's method. It is only applicable to first-order differential equations and may not be accurate for complex functions or large step sizes. It also requires a significant amount of computation, making it time-consuming for more complicated problems.

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