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What is the Taylor's principle of state?

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Solve the following using Taylor’s Series Method (up to 3 terms): \[ \frac{dy}{dx} = x^2 + y, \quad y(0) = 1 \] Find \( y(0.2) \).

Solve the following using Taylor’s Series Method (up to 3 terms): \[ \frac{dy}{dx} = x^2 + y, \quad y(0) = 1 \] Find \( y(0.2) \).
Answer : #### **Solution:** 1. **Taylor Series Expansion:** The Taylor series expansion of \( y(x) \) around \( x = 0 \) is: \[ y(x) = y(0) + x \frac{dy}{dx} \bigg|_{x=0} + \frac{x^2}{2!} \frac{d^2y}{dx^2} \ ... + 0.004 \] \[ = 1.224 \] **Final Answer: \( y(0.2) \approx 1.224 \)**. ---...

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