Question: Solve $\frac{dy}{dx} = xy$ with the help of Eulers method, given that y(0) = 1 and find y when x = 0.3 (h = 0.1)

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Solution:

$\begin{aligned} \frac{d y}{d x}=x . y=f(x, y) & x_{0}=0, y_{0}=1 \\ y_{n}=y_{n-1}+h . f\left(x_{n-1}, y_{n-1}\right) \end{aligned}$

$\begin{array}{|c|c|c|c|c|}\hline \text { Iteration (n) } & {x_{n}} & {y_{n}} & {f\left(x_{n}, y_{n}\right)} & {y_{n+1}=y_{n}+h.f(x_{n},y_{n})} \\ \hline 0 & {0} & {1} & {0} & {1} \\ \hline 1 & {0} & {1} & {0} & {1,01} \\ \hline 2 & {0.2} & {1.01} & {0.202} & {1.0302} \\ \hline\end{array}$

$\therefore y(0.3)=1.0302$

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