While studying linear algebra I encountered the following result regarding the solutions of a non homogeneous system of linear equations: if S o l ( A , b b ) ≠ ∅ S o l ( A , b b ) = x x + S o l ( A , 0 0 )Where x is a particular solution.Then, while studying differential equations, I found that the solutions for y ′ = a ( x ) y + b ( x )are all of the form y = y p + C e A ( x ) Where the last term refers to the solutions of the associate homogeneous equation and y p is a particular solution.These two concepts seem related to me, but I've not been able to find/think about a satisfying reason, so I'd like to know whether there is actually a relation or not.I should specify that I've not been able to study properly systems of differential equations of the first order or equations of the n-th order yet, I suspect that the answer is there but I can't be sure. However I'd like to get the complete answer, so if it involves the latter topics it's not a problem.

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While studying linear algebra I encountered the following result regarding the solutions of a non homogeneous system of linear equations: if   S  o  l  (  A  ,            b              b    )  ≠  ∅  S  o  l  (  A  ,            b              b    )  =            x              x    +  S  o  l  (  A  ,            0              0    )Where x is a particular solution.Then, while studying differential equations, I found that the solutions for      y    ′    =  a  (  x  )  y  +  b  (  x  )are all of the form  y  =      y    p    +  C      e          A      (      x      )      Where the last term refers to the solutions of the associate homogeneous equation and       y    p   is a particular solution.These two concepts seem related to me, but I&#039;ve not been able to find/think about a satisfying reason, so I&#039;d like to know whether there is actually a relation or not.I should specify that I&#039;ve not been able to study properly systems of differential equations of the first order or equations of the n-th order yet, I suspect that the answer is there but I can&#039;t be sure. However I&#039;d like to get the complete answer, so if it involves the latter topics it&#039;s not a problem.

nicoupsqb · Accepted Answer

This can be useful to visualize the correlation between linear algebra and systems of differential equations:If   A and   B are two linear spaces, let   Γ  :  A  →  B be a linear operator with   b  ∈  B, then the level set of   b of    Γ can be obtained translating   K  e  r  (  Γ  ) of a factor       Γ          −      1        (  b  ), hence  ∀  γ  ∈      Γ          −      1        (  b  )  :      Γ          −      1        (  b  )  =  γ  +  K  e  r  (  Γ  )  .Let&#039;s now consider   y  ∈            C              (      n      )        (  I  ) and define      Γ                  a        0            ,      …      ,              a                  n          −          1                      (  y  )  :=      y          (      n      )        +            ∑              i        =        0                    n      −      1            a    i    (  x  )      y          (      i      )        .The derivative defines a linear operation, so   Γ is a linear operator such that   Γ  :            C              (      n      )        (  I  )  →            C              (      0      )        (  I  ).Now solving the system   (  ∗  )      {                            y          (                      x            0                    )          =                      y            0                                                ⋮                                                  y                          (              n              −              1              )                                (                      x            0                    )          =                      y            0                          (              n              −              1              )                                               is equivalent to   Γ  (  y  )  =  b, and solving the homogeneous equation associated to   (  ∗  ) is equivalent to   Γ  (  y  )  =  0.We have showed that the the general integral of   (  ∗  ) is a linear variety and for this reason it can be expressed by the sum of a particular solution of   (  ∗  ) with a solution of the associated homogeneous equation.

While studying linear algebra I encountered the following result regarding the solutions of a non ho

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