Problem 3\ Set

s=s(t)=lnt

, i.e.,

t=e^(s)

. Rewrite the ODE, for

x=x(t)

,\

t^(2)x^()+atx^()+bx=0 in (0,\\\\infty )

\ as an ODE where the independent variable is

s

rather than

t

, i.e., as an ODE for\

y=y(s)

in

R

. Hint: Use the chain rule on

y(s(t))=x(t)

.

Set s=s(t)=lnt, i.e., t=es. Rewrite the ODE, for x=x(t), t2x+atx+bx=0in(0,) as an ODE where the independent variable is s rather than t, i.e., as an ODE for y=y(s) in R. Hint: Use the chain rule on y(s(t))=x(t)