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We are asked to solve the differential equation numerically using the Taylor series method of order $O(h^2)$ and compare with the exact solution $y = -\frac{1}{x}$. Let’s solve it step by step manually. The differential equation is: $$ x^2 y' = 1 - xy - x^2 y^2, \quad y(1) = -1 $$ or equivalently, $$ y' = \frac{1 - xy - x^2 y^2}{x^2}. $$ The Taylor series method of order 2 formula is: $$ y_{n+1} = y_n + h f(x_n, y_n) + \frac{h^2}{2} f_x(x_n, y_n) + \frac{h^2}{2} f_y(x_n, y_n) f(x_n, y_n), $$ where $f(x, y) = y'$, and $f_x = \frac{\partial f}{\partial x}, f_y = \frac{\partial f}{\partial y}$. --- ### Step 1: Compute derivatives of $f(x, y)$ $$ f(x, y) = \frac{1 - xy - x^2 y^2}{x^2} = \frac{1}{x^2} - \frac{y}{x} - y^2 $$ * Partial derivative w\.r.t $x$: $$ f_x = \frac{\partial}{\partial x} \left(\frac{1}{x^2} - \frac{y}{x} - y^2 \right) = -\frac{2}{x^3} + \frac{y}{x^2} \quad (\text{treat } y \text{ as constant}) $$ * Partial derivative w\.r.t $y$: $$ f_y = \frac{\partial}{\partial y} \left(\frac{1}{x^2} - \frac{y}{x} - y^2 \right) = -\frac{1}{x} - 2y $$ --- ### Step 2: Taylor series formula simplified $$ y_{n+1} = y_n + h f(x_n, y_n) + ________ ________ _______ _____ _____ _______ _______ _______ ______ ___.
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