Advanced Engineering Mathematics: Euler Method

In summary, to solve the given problem of y(prime) = (y-x)^2 with initial condition y(0) = 0 and step size h = 0.1, one can use Euler's method or Taylor series. However, the exact solution is not obvious since the equation is not linear. To get a bound on the error, one can think about the problem in terms of Taylor series and compare the difference between using Euler's method and a Taylor series. It may also be helpful to rewrite the equation in terms of u and consider u' as a hint for solving the equation.
  • #1
think4432
62
0
Do 10 steps. Solve the problem exactly. Compute the error (Show all details).

The problems says do 10 steps, but 3-4 steps will suffice!

Problem: y(prime) = (y-x)^2
y(0) = 0
h = 0.1

I don't understand how to get the exact solution and what to do from there!
I know that,
f(x,y) = (y-x)^2

And that u = (y-x)

But from there, I am stuck!

Help!
 
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  • #2
The differential equation can be solved. However it's not a very obvious solution since it is not linear.

However you can get a bound on the error if you think about the problem in terms of Taylor series. Specifically what's the difference in using Euler's method versus a Taylor series?
 
  • #3
Feldoh said:
The differential equation can be solved. However it's not a very obvious solution since it is not linear.

However you can get a bound on the error if you think about the problem in terms of Taylor series. Specifically what's the difference in using Euler's method versus a Taylor series?

I don't think we're learning about Taylor series, but I just don't understand how we would solve the DE...

I can probably apply to Euler's method after solving it...
 
  • #4
Hint: What is u' equal to? Rewrite the original differential equation in terms of u.
 

1. What is the Euler Method?

The Euler Method is a numerical method used to approximate solutions to ordinary differential equations (ODEs). It is based on the idea of breaking down a continuous function into small, discrete steps and using the slope of the function at each step to estimate the value at the next step.

2. How does the Euler Method work?

The Euler Method works by using the derivative of a function to estimate the value of the function at a particular point. It takes an initial value and an increment value, and then uses these to calculate the next point on the function. This process is repeated until the desired accuracy is achieved.

3. What types of problems can be solved using the Euler Method?

The Euler Method can be used to solve a wide range of problems that can be represented by ordinary differential equations. This includes problems in physics, engineering, and other scientific fields where a system's behavior can be described by a differential equation.

4. What are the advantages of using the Euler Method?

One of the main advantages of the Euler Method is its simplicity. It is relatively easy to implement and does not require extensive computational resources. Additionally, it can provide a good approximation of the solution to a differential equation, especially when the step size is small.

5. What are the limitations of the Euler Method?

The Euler Method has several limitations, including being prone to error when the step size is too large. It also does not always provide accurate results for complex systems with rapidly changing behavior. Additionally, it can only be used for first-order differential equations, limiting its applicability in some cases.

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