Answer to Question #308245 in Differential Equations for Junior Pyakara

Question

Find the constants C0, C1, and x1 so that the quadrature formulae z0^1 f(x) dx = C0 f(0) + C1 f(x1),

Has the highest possible degree of precision

Answer 1

Due to the fact that is the boundary point of the segment, the highest possible degree of precision is $2$ (number of nodes minus $2$). Solve the equation for $f(x)=ax^2+bx+c$ :

$\int_0^1ax^2+bx+c=c_0f(0)+c_1f(x_1)\\ \frac{a}{3}+\frac{b}{2}+c=c_0c+c_1ax_1^2+c_1bx_1+c_1x$

Due to the fact that $a, b, c$ are arbitrary numbers, the following system of equations is obtained (grouping the terms by $a, b, c$) :

$\begin{cases} \frac{1}{3}=c_1x_1^2\\ \frac{1}{2}=c_1x_1\\ 1=c_0+c_1 \end{cases}$

Solving the system we get the answer: $x_1=\frac{2}{3},\ c_1=\frac{3}{4},\ c_0=\frac{1}{4}$