Please leave a hint and your idea not the whole solutions. Every idea is accepted. Please let me know what you think.
The Problem: Let we have the following Legendre differential equation $$(1-t^{2})\frac{d^{2}u}{dt^{2}}-2t\frac{du}{dt}+(\chi-c^{2}t^{2})u=0.$$ Assume that $u$ is an entire function. I can see that if $u(1)=0$ then $u'(1)=0$ and also other order of the differention will be zero, i.e. $u^{(n)}(1)=0.$
Now my problem is that how can I see that if $u(1)=0$ then $u(t)=0.$