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Let $\phi_{0}(t)=0$ and use the method of successive approximations to solve the given initial value problem. (a) Determine $\phi_{n}(t)$ for an arbitrary value of ...

Question

Let $\phi_{0}(t)=0$ and use the method of successive approximations to solve the given initial value problem. (a) Determine $\phi_{n}(t)$ for an arbitrary value of $n .$ (b) Plot $\phi_{n}(t)$ for $n=1, \ldots, 4$. Observe whether the iterates appear to be converging. $$y^{\prime}=t y+1, \quad y(0)=0$$

let $\phi_{0}(t)=0$ and use the method of successive approximations to solve the given initial value problem. (a) Determine $\phi_{n}(t)$ for an arbitrary value of $n .$ (b) Plot $\phi_{n}(t)$ for $n=1, \ldots, 4$. Observe whether the iterates appear to be converging. $$y^{\prime}=t y+1, \quad y(0)=0$$



Answers

Determine the recursive formulas for the Taylor method
of order 2 for the initial value problem
$$ y^{\prime}=x y-y^{2}, \quad y(0)=-1 $$


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