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To solve the initial value problem \( y' = x - y \) with \( y(0) = 1 \) at \( x = 0.1 \) using the third-order Taylor’s series method, we follow these steps: --- ### Step 1: Write the Taylor series expansion up to the third order The Taylor series expansion for \( y(x + h) \) is: \[ y(x + h) = y(x) + h y'(x) + \frac{h^2}{2!} y''(x) + \frac{h^3}{3!} y'''(x) + \cdots \] We need to compute \( y(0.1) \) using \( h = 0.1 _________ ____ _________ _____ ______ ____ ___ ___ ______ ________ ___.
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