Is the Advanced Euler Method Accurate for Solving Differential Equations?

[t_n_p]

Homework Statement

integrate dy/dx = (1+e^y)(1+x) from x=0 using the improved euler method taking step sizes of 0.125. compare the numerical solution of y(0.5) with the exact value

The Attempt at a Solution

Firstly to find the exact value I use separation of variables and the initial condition y(0)=0 to find that y=-ln[-1+2e^(-x)e^{-(x^2)/2}]

Subbing in x=0.5 yields 2.6518

However, using an advanced euler calculator i found online tells me this:

x_n y_n
0.00000 0.00000
0.12500 0.28560
0.25000 0.67185
0.37500 1.25597
0.50000 2.45219

That is, y(0.5)=2.45219

I was just wondering if somebody would be able to confirm this is correct.

Thanks!

 

[mighty2000]

Your approach and calculations seem to be correct. The advanced Euler method is a numerical approximation technique and therefore may not give the exact solution. However, the value of y(0.5) obtained using the improved Euler method with step size 0.125 is very close to the exact value of 2.6518. This shows that the method is effective in approximating the solution to the given differential equation.

To confirm the accuracy of your results, you can also try using smaller step sizes (e.g. 0.0625, 0.03125) and see if the value of y(0.5) gets closer to the exact value. This is because smaller step sizes result in a more accurate approximation.

In conclusion, your calculations and results seem to be correct. Keep up the good work!