Question

# Transform the second-order differential equation \frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=x into a system of first-order differential equations.

Differential equations
Transform the second-order differential equation $$\displaystyle{\frac{{{d}^{{{2}}}{x}}}{{{d}{t}^{{{2}}}}}}+{\frac{{{d}{x}}}{{{d}{t}}}}={x}$$ into a system of first-order differential equations.

2021-06-06

Let $$\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={v}$$ (1), then we have
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}\right)}+{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={x}$$
$$\displaystyle\Rightarrow{\frac{{{d}{v}}}{{{\left.{d}{t}\right.}}}}+{v}={x}$$ (2)
From (1) and (2), we have the system
$$\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={0}{x}+{v}$$
$$\displaystyle{\frac{{{d}{v}}}{{{\left.{d}{t}\right.}}}}={x}-{v}$$
$$\Rightarrow \begin{bmatrix}\frac{dx}{dt} \\ \frac{dv}{dt} \end{bmatrix}=\begin{bmatrix}0 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ v \end{bmatrix}$$