Writing a system of 2 ODEs as a 1st order ODE

[Ted123]

Homework Statement

Consider the following initial value problem for two functions $y(x),z(x)$: $$0 = y''+(y'+7y)\text{arctan}(z)$$ $$5z' = x^2+y^2+z^2$$ where $0 \leqslant x \leqslant 2,\; y(0)=1.8,\;y'(0)=-2.6,\;z(0)=0.7$.

Rewrite the system of ODEs in standard form using a suitable substitution.

The Attempt at a Solution

Would putting ${\bf u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}$ where $u_1(x) = y(x),\; u_2(x) = y'(x),\; u_3(x)=z(x)$ work?

Then:

$$u_1' = y' = u_2$$ $$u_2' = y'' = -(y'+7y)\arctan(z) = -(u_2+7u_1)\arctan(u_3)$$ $$u_3' = z = \frac{1}{5} ( x^2 + u_1^2 + u_3^2)$$

so that ${\bf u'} = \begin{bmatrix} u_1' \\ u_2' \\ u_3' \end{bmatrix} = \begin{bmatrix} u_2 \\ -(u_2+7u_1)\arctan(u_3) \\ \frac{1}{5} ( x^2 + u_1^2 + u_3^2) \end{bmatrix} ,\; 0 \leqslant x \leqslant 2$ and ${\bf u}(0) = \begin{bmatrix} 1.8 \\ -2.6 \\ 0.7 \end{bmatrix}$

 

[HallsofIvy]

Yes, that is perfectly correct. Note that this does NOT reflect your title! You are NOT "Writing a system of 2 ODEs as a 1st order ODE". You are, rather, writing a higher order system of equations as a system of first order ODEs.