Chapter 5.3, Problem 50E

(a)

To determine

The Legendre polynomials $P_6\left(x\right)\text{ and }{P}_7\left(x\right)$ using the explicit solutions $y_1\left(x\right)\text{ and }{y}_2\left(x\right)$ of Legendre’s equation $\left(1-x^2\right)y^{″}-2x{y}^{\prime }+n\left(n+1\right)y=0$ and by making appropriate choice of $c_0\text{ and }{c}_1$.

(b)

To determine

The differential equations for which $P_6\left(x\right)\text{ and }{P}_7\left(x\right)$ are particular solutions.