Calculation using Euler's method

Question

Given y' = 1 - 2x - 3y, starting condition y(4)=5 and h = 1/2

I am asked to estimate by hand the value for y(5).

My question is, if my staring value are as follows:

start:    x=4     y=5     y'= -22

then I suppose I have to do the Euler method the other way around and double the x-value instead of halving it since I need to get to $x=5$. However, in that case the next x-value will be $x=8$. How do I get $x=5$ if my step length is $1/2$?

Sorry if this question seems dumb or easy but I really don't see how I can get to the answer.

Answer 1

You compute in two steps first an approximation to $y(4.5)=y(4+0.5)$ and with that then $y(5)=y(4.5+0.5)$, all using the formula $$y(x+h)\approx y(x)+h\,f(x,y(x))$$ of the Euler forward method.

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Setting $u=3y+2x-1$ so that $y'=-u$ gives $u'=3y'+2=-3u+2$. Then setting $v=3u-2$ so that $u'=-v$ results in $v'=3u'=-3v$, so that $v=Ce^{-3x}$. Inserting backwards gets $u=\frac23+Ce^{-3x}$, so that the exact solution is $$y(x)=\frac19-\frac23x+Ce^{-3x},$$ the constant $C$ is determined by the initial condition.

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Of course you need $Lh<0.1$, where $L$ is the $y$-Lipschitz constant, here $L=3$, to get at least one digit correct. So usable step sizes are $h=0.03$ and below.

In python you get values for finer step sizes with the script

for N in [10,20,30,100]:
    x, y, h = 4, 5, 0.5/N;
    print "\nN=%3d, h=%.3g\n----"%(N,h)
    for k in range(3):
        print "%3d:  %8.4f  %10.6f"%(k,x,y);
        if k==3: continue
        for j in range(N): x,y = x+h, y+h*(1-2*x-3*y);

producing the table

N= 10, h=0.05
----
  0:    4.0000    5.000000
  1:    4.5000   -1.044449
  2:    5.0000   -2.502154

N= 20, h=0.025
----
  0:    4.0000    5.000000
  1:    4.5000   -0.948994
  2:    5.0000   -2.463288

N= 30, h=0.0167
----
  0:    4.0000    5.000000
  1:    4.5000   -0.918124
  2:    5.0000   -2.450170

N=100, h=0.005
----
  0:    4.0000    5.000000
  1:    4.5000   -0.875670
  2:    5.0000   -2.431692