Get first order derivative from second order derivative

Question

I am given:

$$ y'' = 11-y $$

$$y(2)=1 ; y'(2)=-4$$

and asked to use Euler's method to find $y(2.2)$ for $h=0.1$

To find $y'$ I simply took the integral of $y''$ to get:

$$y'=11x-yx$$

However, this does not satisfy the condition given above, that $y'(2)=-4$. Is this not the correct way of obtaining the first order derivative?

Answer 1

The first-order system for your second-order equation is simply \begin{align} y'&=v,\\ v'&=11-y, \end{align} and the first Euler step correspondingly \begin{align} y_1&=y_0+hv_0,\\ v_1&=v_0+h(11-y_0). \end{align}