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## Homework Statement

Solve the differential equation

[tex]\frac{dy^2}{dx^2}=xy^2-2yy'+x^3+4[/tex]

where

[tex]y(1)=1[/tex]

[tex]y'(1)=2[/tex]

by means of the Taylor-series expansion to get the value of y at x=1.1. Use terms up to [tex]x^6[/tex] and [tex]\Delta x=0.1[/tex]

## The Attempt at a Solution

I'm unsure as to how I should go about determining the coefficients for the Taylor expansion from the given equation. What I mean is that I don't know how to apply implicit differentiation techniques to higher order derivatives as in this case. Do we treat the derivatives in the equation as if they were variables? In other words, should I differentiate as follows (I'm using brackets to easier relate the the original term above to my subsequent derivative of said term):

[tex]\frac{dy^3}{dx^3}=(y^2+2xyy')-(2y'+2yy'')+3x^2[/tex]

?

Perhaps there's even an easier way that I'm missing?

Any help will be greatly appreciated.

phyz

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