Verify that y = sin ⁡ x cos ⁡ x − cos ⁡ x is a solution of the initial value problem y ′ + ( tan ⁡ x ) y = c o s 2 x y ( 0 ) = − 1 on the interval − π / 2 &lt; x &lt; π / 2

Question

Verify that   y  =  sin  ⁡  x  cos  ⁡  x  −  cos  ⁡  x is a solution of the initial value problem       y    ′    +  (  tan  ⁡  x  )  y  =  c  o      s          2        x  y  (  0  )  =  −  1 on the interval   −  π      /    2  &amp;lt;  x  &amp;lt;  π      /    2

slapadabassyc · Accepted Answer

y&#039;+y P(x)= Q(x)      y    ′    +  (  tan  ⁡  x  )  y  =      cos          2        ⁡  x linear diff. eqn.  P  (  x  )  =  tan  ⁡  x  Q  (  x  )  =      cos          2        ⁡  xintegration factor   i  (  x  )  =  exp  ⁡  (  ∫  P  (  x  )  d  x  )  =  exp  ⁡  (  ∫  tan  ⁡  (  x  )  d  x  )  =  exp  ⁡  (  −  ln  ⁡  (  cos  ⁡  x  )  =      1          cos      ⁡      x        y  .  i  (  x  )  =  ∫  i  (  x  )  Q  (  x  )  d  x  +  c  y  ∗      1          cos      ⁡      x        =  ∫      1          cos      ⁡      x            cos          2        ⁡  x  d  x  +  c  y  ∗      1          cos      ⁡      x        =  sin  ⁡  x  +  c  y  =  sin  ⁡  x  .  cos  ⁡  x  +  c  .  cos  ⁡  x  y  (  0  )  =  −  1  ⇒  −  1  =  sin  ⁡  0  cos  ⁡  0  +  c  .  c  o  s  0  ⇒  c  =  −  1  y  =  sin  ⁡  x  .  cos  ⁡  x  −  cos  ⁡  x

Verify that y=sin xcos x-cos x is a solution of the initial value problem y'+(tan x)y=cos^2 xy(0)=-1 on the interval -pi /2 < x< pi/2

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New Questions in Calculus 1