Ordinary differential equations, please solve this......

[akshatha]

1)
If { Ø1(x), Ø2(x) } is one set of two linearly independent solution on a ? x ? b and { ?1(x), ?2(x) } be another set of two linearly independent solution, then show that there exists a constant c ? 0 such that
W[ ?1(x), ?2(x) ] = c[ Ø1, Ø2 ](x) for all a ? x ? b.
where W[] is wronskian.

2)
If one of the solution of the linear equation
(1-x) y'' + xy' - y=0,
is y1(x)=x; find the second solution y2(x) such that y1(x) and y2(x) are linearly independent on some interval L.

 

[Deleted member 4993]

1)
If { Ø1(x), Ø2(x) } is one set of two linearly independent solution on a ? x ? b and { ?1(x), ?2(x) } be another set of two linearly independent solution, then show that there exists a constant c ? 0 such that
W[ ?1(x), ?2(x) ] = c[ Ø1, Ø2 ](x) for all a ? x ? b.
where W[] is wronskian.

2)
If one of the solution of the linear equation
(1-x) y'' + xy' - y=0,
is y1(x)=x; find the second solution y2(x) such that y1(x) and y2(x) are linearly independent on some interval L.

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

Think about applying the lema proven in (1) into (2).

What is the expression for Wronskian?