Find the fourth degree Taylor polynomial of the solution to the differential equation

The initial conditions yield *a*_{0}=1 and *a*_{1}=0. Since *y*''=-*y*^{2},
we obtain that

Differentiating *y*''=-*y*^{2} yields

and hence *a*_{3}=0.

When we differentiate

we obtain

and consequently

The fourth degree Taylor approximation has the form

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*Helmut Knaust
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*1998-06-29*
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