Find a degree 3 Taylor polynomial approximation to exyysinx

Find a degree 3 Taylor polynomial approximation to e^xy\'\'+y=sin(x) ; y(0)=1, y\'(0)=1

Solution

f = e^(xy\"+y) sin(x)-----> f(0,0) = 1

we can difertiate both sides,then
d (e^(x,y^\"+y) )=d(sinx)

e^xy2.d(xy^2)+y\'=cosx

e^{xy^2}(y^2+x.2y.y\')+y\'=cosx

y^2e^{xy^2}+2xy e^{xy^2}.y\'+y\'=cosx

-y^2e^{xy^2} +y\'=-y^2e^{xy^2}

==>(2xy e^{xy^2}.+1)y\'=cosx-y^2 e^xy^2

therfore y\'=cosx-y^2 e^xy^2/2xy e^xy^2+1