We seek to find the path \(y(x)\) that minimizes the integral \(I=\int f\left(x, y, y^{\prime}ight) d x\). Find Euler's equation for \(y(x)\) for each of the following integrands \(f\), and then find the solutions \(y(x)\) of each of the resulting differential equations if the two endpoints are \((x, y)\) \(=(0,1)\) and \((1,3)\) in each case.

(a) \(f=a x+b y+c y^{\prime 2}\)

(b) \(f=a x^{2}+b y^{2}+c y^{\prime 2}\).