Questions

01

Variation of parameters: triple repeated root

Suppose $y^{\prime \prime \prime }+b{y}^{\prime \prime }+c{y}^{\prime }+dy=0$, and suppose $r^3+b{r}^2+cr+d=(r-\lambda {)}^3$ for some $\lambda \in \mathbb{R}$. Use variation of parameters to find two (independent) additional solutions besides $e^{\lambda t}$.

02

Variation of parameters: Legendre second solution

The Legendre equation for $p=1$ is $(1-x^2)y^{\prime \prime }-2x{y}^{\prime }+2y=0$. One solution is $y_1=x$. Find a second solution.