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Chapter 4: Q44E (page 200)

Find a second linearly independent solution using reduction of order.

ty''+(1-2t)y'+(t-1)y=0,t>0;f(t)=et

Short Answer

Expert verified

The second linearly independent solution of the given equation:

ty''+(1-2t)y'+(t-1)y=0,t>0;f(t)=etis y=c1et+c2etlnt.

Step by step solution

01

Finding y

Given differential equation isty''+(1-2t)y'+(t-1)y=0 then the standard form of the solution isy''+1-2tty'+t-1tx=0 andf(t)=et is one of the solutions and p(t)=1-2tt.

Then:

y2=y1(t)鈭�e-鈭�p(t)dty12(t)dt.

02

Simplification

Simplify the above

dy=et鈭�e-鈭�1-2ttdtet2dt=et鈭�e2t-ln(t)e2tdt=et脳鈭�t-1脳e2te2tdty=lntet

So, the solution is y=c1et+c2etlnt.

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