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zabumafu
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Homework Statement
t^2y''-4ty'+6y=0 t>0, y1(t)=t^2 find the 2nd solution using method of reduction of order.
Homework Equations
The Attempt at a Solution
Let y=v(t)t^2
y'=2tv+v't^2
y''=2v+2tv'+v''t^2+2tv'
put in back into eq and solving reduces to (checked it twice)
t^4*v''=0
v''=0
Then I integrated with respect to T
v'=c (c is some constant)
v(t)=ct where c is a constant.
I substituted v(t) back into y=v(t)*t^2 and got y2=ct^3 where c is some constant so y2=t^3? I know that's the answer but did I correctly do all these steps? Also do I need to show the wronskian in order to prove that the solution is acceptable as y1 and y2 need to form a fundamental set of solutions? Thanks I just want to make sure I am doing this correctly.
--------Problem 2: same question different equation
(x-1)y''-xy'+y=0 x>1 y1=e^x
let y=v(t)e^x
y'=v'e^x +ve^x
y''=v''e^x +2v'e^x +ve^x
into the eq:
e^x[(x-1)(v''+2v'+v)-x(v'+v)+v]=0
(x-1)v''+(x-2)v'=0 then I get confused
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