diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md index a3e1fff86b91386b4c8be9ec6a8a0bcceefd27d5..dbeafe893ed9d0838f0d2c00e43c80db07a423bc 100644 --- a/src/8_differential_equations_2.md +++ b/src/8_differential_equations_2.md @@ -55,11 +55,13 @@ $$y_{1} = y, \ y_{2} = y', \ \cdots, \ y_{n} = y^{(n-1)}.$$ Then, the differential equation can be re-written as -$$y_1 ' = y_2$$ -$$y_2 ' = y_3$$ -$$ \vdots $$ -$$y_{n-1} ' = y_{n}$$ -$$y_{n} ' = - a_{0} y_{1} - a_{1} y_{2} - \cdots - a_{n-1} y_{n}.$$ +$$\begin{split} +y_1 ' = & y_2 \\ +y_2 ' = & y_3 \\ +\vdots & \\ +y_{n-1} ' = & y_{n} \\ +$$y_{n} ' = & - a_{0} y_{1} - a_{1} y_{2} - \cdots - a_{n-1} y_{n}. +\end{split}$$ Notice that together these $n$ equations form a linear first order system, the first $n-1$ equations of which are trivial. Note that this trick can be used to