# Don't understand x''=x^2-y-xy,\ \ \ y''=y-x as a system of first order

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Lakisha Archer

1. Let, then the $x{}^{″}={x}^{2}-y-xy$ equation converted to ${x}_{2}^{\prime }={x}_{1}^{2}-y-{x}_{1}y$ where ${x}_{1}^{\prime }={x}_{2}$.
Let,
${y}_{1}=y,{y}_{2}={y}^{\prime }$ then the equation ${y}^{″}=y-x$ converted to ${y}_{2}^{\prime }={y}_{1}-x$  where ${y}_{2}={y}_{1}^{\prime }$.
2. $x{}^{‴}=tx$ is already a non-autonomous system. We need to convert it to first order equation.
Let, , the above system transformed to ${x}_{3}^{\prime }=t{x}_{1}$ which is a non-autonomous system of first order differential equation.
3. $x{}^{⁗}=tx$ is not a autonomous system of linear equation. First transfer it to autonomous.
Put, $x=tv$.
The equation reduces to $\frac{3{d}^{2}v}{{dt}^{2}}+\frac{x}{v{d}^{3}}\frac{v}{{dt}^{3}}=\frac{{x}^{2}}{v}wheret=\frac{x}{v}$.
Now let, then the above equation reduces to which is an autonomous system of first order differential equation.