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The problem I have is the initial value problem $$y''' = x + y$$ with $$ y(1) = 3, y'(1) = 2, y''(1) = 1$$ that should be solved with Eulers method using the step length, $h = \frac{1}{2}$.

The iteration step for Eulers method is $y_{n+1} = y_n + hf(x_n, y_n)$. So I should need a system of equations of my initial values I have. I started with substituting: $$U_1 = y, U_2 = y', U_3 = y''$$ and then $$U_1' = U_2, U_2' = U_3, U_3' = x + U_1$$ And it's here I'm stuck, I don't understand how I should start iterate with the step-length from here, I have only encountered first-order problem with Eulers method so I would love if someone could point me in the right direction.

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$y'''(1)=y(1)+(x=1)$ To get it going.

$y(1+1h)=y(1)+y'(1)h$

$y'(1+1h)=y'(1)+y''(1)h$

$y''(1+1h)=y''(1)+y'''(1)$

$y'''(1+1h)=y(1)+(x=1)$

repeat from step 2

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