The Runge-Kuttamethodsare an importantfamilyof implicitand explicititerativemethodsfor the approxi-
mationof solutionsof our ODE.On them,applySimpson'srule:
.
Exercise1.Applythe Euler's,improvedEuler'sand the Runge-Kuttamethodson the ODE
to approximatethe solutionthat satisfyy(0) = 1 fromx= 0 tox= 1 withh= 0.2.We knowthe exactsolutionisy=ex. Comparetheir relativeaccuracyagainstthe exactsolution.