For a differential equation say y′=ty3 with y(0)=1 how would we know if the sequence from Euler's method exists or not?

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For a differential equation say y′=ty3 with y(0)=1 how would we know if the sequence from Euler&#039;s method exists or not?

llevochalecoiozq · Accepted Answer

Step 1The Euler method for this example reads asyn+1=yn+htnyn3.According to the nature of the ODE, consider yn−2. Then the iteration is approximatelyyn+1−2=yn−2(1+htnyn2)−2=yn−2(1−2htnyn2+3h2tn2yn4+O(h3))≈yn−2−2htn,as long as hyn2 remains small, yn−2ggh. As tn=nh this can be solved asyn−2≈y0−2−n(n−1)h2=y0−2−tn(tn−h).This means that at tn≈1, n=⌊1h⌋, the Euler iteration will reach a value yn≈1h, and thus yn+1≈2h, and from that moment on the second term will dominate,hyn+k+1≥(hyn+k)3⇒yn+k≥1h23k−1,k≥1Normal 64bit floating point numbers go up to 21024, that range gets exceeded before k=7While in theory the Euler sequence can be indefinitely prolonged, in praxis there is no computer that can do this, the (observable, estimated) universe gets exhausted before k=200

For a differential equation say \(\displaystyle{y}'={t}{y}^{{{3}}}\) with

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