differential equations of second orderHow may I solve this differential equations: y ″ + 4 y = 12 x 2 − 16 x cos ⁡ ( 2 x ) ?

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differential equations of second orderHow may I solve this differential equations:      y    ″    +  4  y  =  12      x    2    −  16  x  cos  ⁡  (  2  x  )  ?

empatiji2v · Accepted Answer

Step 1Solve the homogeneous part first: the part       y    ″    +  4  y  =  0 by letting   y  =      e          λ      x      . Then differentiate, substitute, simplify and solve.Step 2Then solve the rest using method of undetermined coefficient.

Beckett Henry · Accepted Answer

Step 1The solution of the homogeneous part of the equation:      y    ″    +  4  yLet   y  =      e          m      x      ,       y    ′    =  m      e          m      x      ,       y    ″    =      m    2        e          m      x            m    2    +  4  =  0  m  =  ±  2  iStep 2Complementary function:  y  =  A  cos  ⁡  2  x  +  B  sin  ⁡  2  xYou now need to find the particular integral:Hint: You may want to try the following function:  y  =  A      x    2    +  B  x  +  C  +  (  D      x    2    +  E  x  +  F  )  cos  ⁡  2  x  +  (  G      x    2    +  H  x  +  I  )  sin  ⁡  2  x

Differential equations of second order. How may I solve this differential equations: y′′+4y=12x^2-16x cos (2x)?

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