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23 Nov 2019
Define the Schwarzian derivative of a function v to be
S(v)= (vâââ÷ vâ)- 3/2(vââ÷vâ) ^2
a. Prove that if f and g are both solutions to yââ+q(x)y= 0 then v=f/g is a solution to the third order differential equation S(v)=2q(x)
b. As an interesting follow up suppose again that f and g are bothsolutions to yââ+q(x)y= 0. Letting v= fg, prove that v satisfiesthe second order, non-linear differential equation 2yyââ= (yâ)^2-4y^2q(x)
Define the Schwarzian derivative of a function v to be
S(v)= (vâââ÷ vâ)- 3/2(vââ÷vâ) ^2
a. Prove that if f and g are both solutions to yââ+q(x)y= 0 then v=f/g is a solution to the third order differential equation S(v)=2q(x)
b. As an interesting follow up suppose again that f and g are bothsolutions to yââ+q(x)y= 0. Letting v= fg, prove that v satisfiesthe second order, non-linear differential equation 2yyââ= (yâ)^2-4y^2q(x)
