# Matrix determinants and differentiation

by tickle_monste
Tags: determinants, differentiation, matrix
 HW Helper P: 1,371 t_m: Your system of equations for the derivatives is \begin{align*} \frac 1 u \frac{du}{dx} + \frac{dv}{dx} & = y \\ \frac{du}{dx} + \frac 1 v \frac{dv}{dx} & = 1 \end{align*} Multiply the equations to clear fractions: \begin{align*} \frac{du}{dx} + u \cdot \frac{dv}{dx} & = uy\\ v \cdot \frac{du}{dx} + \frac{dv}{dx} & = v \end{align*} This can be put into matrix form as $$\begin{bmatrix} 1 & u \\ v & 1 \end{bmatrix} \, \begin{bmatrix} {du}/{dx} \\ {dv}/{dx} \end{bmatrix} = \begin{bmatrix} uy \\ v \end{bmatrix}$$ and, as Hurkyl said, Cramer's rule was apparently used. My only reasoning about why it was used in this case: by using Cramer's rule the numerator and denominator of the solutions are easier to keep track of than they are when you use Gaussian elimination.