Discussion
Let \(\dfrac{dy}{dx} = 1 + xy\) with \(y(0) = 2\). Using Picard's method, the first approximation is :
Answer: B
The first approximation is y1 = 2 + x + x2.
Picard's iteration is \(y_{n+1}(x) = y_{0} + \int_{0}^{x} f(t, y_{n}(t))\, dt\), starting from \(y_{0} = 2\). With \(f(x, y) = 1 + xy\):
\[ \begin{aligned} y_{1} &= 2 + \int_{0}^{x} (1 + 2t)\, dt \\ &= 2 + x + x^{2} \end{aligned} \]
Leaving out the initial value 2 gives \(x + x^{2}\), and dropping the \(xy\) term gives \(2 + x\). The constant 20 has no source in the problem.
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