We know that the coefficients of the Taylor series are given by
and that the initial condition is
![{\displaystyle y(0)={\color {MediumVioletRed}4}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/de0f3cc9cdb97cf9991d1414ae1e958b7bb76d78)
Since
solves the differential equation
, or equivalently,
, we must have
![{\displaystyle y'(0)=0^{2}-2y(0)=0-2\cdot {\color {MediumVioletRed}4}={\color {Chocolate}-8}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/c06b93adf8b12fc12e692a4d4617384e85c9abef)
By differentiating the equation for
(with respect to
), we obtain
, which implies that
![{\displaystyle y''(0)=2\cdot 0-2y'(0)=0-2\cdot {\color {Chocolate}-8}={\color {DarkGreen}16}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/6164e5f11ae826c78899286a13f7c526e7243889)
Differentiating the equation once more, we obtain
, so
![{\displaystyle y'''(0)=2-2y''(0)=2-2\cdot {\color {DarkGreen}16}=-30.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/960e7e3bdb6d157f738b1e224b179bfb58d4e965)
We conclude that
,
,
, and
.