When linear differential equation can also directly be solved by variable separable method?Consider the linear differential equation d y d x + p ( x ) y = q ( x ) . 1. What conditions may be imposed on functions p ( x ) and/or q ( x ) so that it could directly be solved by variable separable method rather than integrating factor e ∫ p ( x ) d x . 2. In the case if it can be solved by both variable separable and integrating factor method, which would be more suitable?

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When linear differential equation can also directly be solved by variable separable method?Consider the linear differential equation             d      y              d      x        +  p  (  x  )  y  =  q  (  x  )  . 1. What conditions may be imposed on functions   p  (  x  ) and/or   q  (  x  ) so that it could directly be solved by variable separable method rather than integrating factor       e          ∫      p      (      x      )      d      x      . 2. In the case if it can be solved by both variable separable and integrating factor method, which would be more suitable?

Regan Holloway · Accepted Answer

Comparing the general linear ODE and the separable linear ODE,            d      y              d      x        +  p  (  x  )  y  =  q  (  x  )        and                d      y              d      x        =  f  (  x  )  (  a  y  +  b  )  ,we note that the linear ODE is separable if and only if  q  (  x  )  −  p  (  x  )  y  =  f  (  x  )  (  a  y  +  b  )  ,with a new function f. This is the case if   q  (  x  )  =  c  p  (  x  ) for some constant   c  ∈      R  , since then we obtain   q  (  x  )  −  p  (  x  )  y  =  p  (  x  )  (  c  −  y  )via separation of variables or via an integrating factor is a matter of preference, and with both methods we obtain the general solution  y  (  x  )  =  c  +  C      e          −      P      (      x      )        ,    C  ∈      R    ,where P is an antiderivative of p.

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