Find the solution of the given initial value problem. t y ′ + 2 y = sin ⁡ t , y ( 3.14 / 2 ) = 1 , t &gt; 0

Question

Find the solution of the given initial value problem.  t      y    ′    +  2  y  =  sin  ⁡  t  ,  y  (  3.14      /    2  )  =  1  ,  t  &amp;gt;  0

Bubbinis4 · Accepted Answer

Given equation is   t      y          1        +  2  y  =  sin  ⁡  tDividing by t, we get      y          1        +      2    t    y  =            sin      ⁡      t        t  This is a linear equation of first order in ywhere   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    )  =  1so       1          t              2              +      C          t              2              =  1  ⇒  C  +  1  =      t          2        ⇒  C  =      t          2        −  1Therefore the general solution of (1) is   y  =            sin      ⁡      t              t              2              −            cos      ⁡      t        t    +                    t                  2                    −      1              t              2

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

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