Solving the IVP ty′′-ty′+y=1 where y(0)=0 and y′(0)=2

Widersinnby7 2022-11-20 Answered
Solving the IVP
t y t y + y = 1
where y ( 0 ) = 0 and y ( 0 ) = 2
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Answers (1)

grizintimbp
Answered 2022-11-21 Author has 16 answers
t y t y + y = 1
( s 2 L ( y ) s y ( 0 ) y ( 0 ) ) + ( s L ( y ) y ( 0 ) ) + L ( y ) = 1 s
( s s 2 ) L ( y ) + ( 2 2 s ) L ( y ) = 1 s
L ( y ) + 2 s L ( y ) = 1 s 2 ( 1 s )
with integrating factor s 2 we have
( s 2 L ( y ) ) = 1 1 s y = L 1 ln ( 1 s ) s 2 + C t = 0 t e x x ( t x ) d x + C t = t 0 t e x x d x e t + 1 + C t = t Ei ( t ) e t + 1 + C t
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