Can Euler's Method Solve y'=y/x with Initial Conditions?

In summary, Euler's method is a numerical method for solving ordinary differential equations (ODEs) by approximating the solution with small steps. The method works by using the derivative of the function to estimate the value at a later point, and its formula is y<sub>n+1</sub> = y<sub>n</sub> + hf(x<sub>n</sub>, y<sub>n</sub>). Its limitations include being a first-order method and not accounting for function curvature. It is best used for simple functions with moderate accuracy needs, but for higher accuracy, other methods such as Runge-Kutta may be better suited.
  • #1
lydia_zhu
1
0
Hi everyone.

I was asked to learn Euler's method by myself but I was really confused with this question. How can I work out this equation by using Eulers' method (which is a method solving first order fifferencial equation with initial conditions)
Given：y'=y/x, y(2)=3, use h=0.2, what is y(2.8) ?

thanks very much!
 
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  • #2
You write
[tex]
\frac{dy}{dx}=\frac{y(x+h)-y(x)}{h}
[/tex]
to obtain:
[tex]
\frac{y(x+h)-y(x)}{h}=y(x)/x
[/tex]
To obtain:
[tex]
y(x+h)=y(x)+\frac{y(x)h}{x}
[/tex]
start with [tex]y(2)[/tex] and increment.
 

1. What is Euler's method?

Euler's method is a numerical method for solving ordinary differential equations (ODEs). It involves approximating the solution to an ODE by using a series of small steps instead of solving it directly.

2. How does Euler's method work?

Euler's method works by using the derivative of the function at a given point to estimate the value of the function at a slightly later point. The smaller the step size, the more accurate the approximation will be.

3. What is the formula for Euler's method?

The formula for Euler's method is:
yn+1 = yn + hf(xn, yn),
where yn is the approximation of the solution at the nth step, h is the step size, and f(xn, yn) is the derivative of the function at the nth step.

4. What are the limitations of Euler's method?

Euler's method is a first-order method, meaning that the error in the approximation is proportional to the step size. This can lead to significant errors if the step size is too large. It also does not take into account the curvature of the function, so it may not accurately capture rapid changes in the solution.

5. When should I use Euler's method?

Euler's method is best used when the function is relatively simple and the desired level of accuracy is not too high. It can also be a good starting point for more complex methods of solving ODEs. However, for highly accurate solutions, other methods such as Runge-Kutta methods may be more appropriate.

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