Find the solution of the given initial value problem. ty'+2y=sin t, y(3.14/2)=1, t>0

Emmanuel Pace 2022-07-26 Answered
Find the solution of the given initial value problem.
t y + 2 y = sin t , y ( 3.14 / 2 ) = 1 , t > 0
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Answers (1)

Bubbinis4
Answered 2022-07-27 Author has 8 answers
Given equation is t y 1 + 2 y = sin t
Dividing by t, we get
y 1 + 2 t y = sin t t
This is a linear equation of first order in y
where P ( t ) = 2 t   a n d   Q ( t ) = sin t t
P ( t ) d t = 2 1 t d t = 2 log t = log t 2
Integrating Factor = e P ( t ) d t = e log t 2 = t 2
Multiplying (1) with t 2 , we get
t 2 y 1 + 2 t y = t sin t ( 2 )
(2) can be rewritten as
d d t [ y t 2 ] = t sin t
d d t [ y t 2 ] d t = t sin t . d t
y t 2 = t ( cos t ) 1 ( cos t ) d t
y t 2 = t cos t + sin t + C
y ( t ) = sin t t 2 cos t t + C t 2
we know that y ( π 2 ) = 1
so 1 t 2 + C t 2 = 1 C + 1 = t 2 C = t 2 1
Therefore the general solution of (1) is y = sin t t 2 cos t t + t 2 1 t 2

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