Differential equations notationI've always wondered why does the differential equation notation for linear equations differ from the standard terminology of vector spaces.We all know that the equation y ″ + p ( x ) y ′ + q ( x ) y = g ( x ) for some function g is called linear and that the associated equation y ″ + p ( x ) y ′ + q ( x ) y = 0 is called homogeneous. But why is that? WHY should mathematicians explicitly cause confusion with the rest of the theory of vector spaces?What I mean by that is : Why not call the equation y ″ + p ( x ) y ′ + q ( x ) y = g ( x ) an affine equation and call y ′ + p ( x ) y ′ + q ( x ) y = 0 a linear equation? Because linear equations (in the sense of differential equations) are not linear in the sense of vector spaces unless they're homogeneous ; and linear equations (in the sense of differential equations) remind me more of a linear system of the form A x = b (which is called an affine equation in vector space theory) than of a linear equation at all.Just so that I made myself clear ; I perfectly know the difference between linear equations in linear algebra and linear equations in differential equations theory ; I'm asking for some reason of "why the name".

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Differential equations notationI&#039;ve always wondered why does the differential equation notation for linear equations differ from the standard terminology of vector spaces.We all know that the equation       y    ″    +  p  (  x  )      y    ′    +  q  (  x  )  y  =  g  (  x  ) for some function g is called linear and that the associated equation       y    ″    +  p  (  x  )      y    ′    +  q  (  x  )  y  =  0 is called homogeneous. But why is that? WHY should mathematicians explicitly cause confusion with the rest of the theory of vector spaces?What I mean by that is : Why not call the equation       y    ″    +  p  (  x  )      y    ′    +  q  (  x  )  y  =  g  (  x  ) an affine equation and call       y    ′    +  p  (  x  )      y    ′    +  q  (  x  )  y  =  0 a linear equation? Because linear equations (in the sense of differential equations) are not linear in the sense of vector spaces unless they&#039;re homogeneous ; and linear equations (in the sense of differential equations) remind me more of a linear system of the form   A  x  =  b (which is called an affine equation in vector space theory) than of a linear equation at all.Just so that I made myself clear ; I perfectly know the difference between linear equations in linear algebra and linear equations in differential equations theory ; I&#039;m asking for some reason of &quot;why the name&quot;.

Clare Acosta · Accepted Answer

Step 1There is no confusion at all: these are in fact the same concepts when viewed in the right light.First of all, we need to find a vector space to put all these functions in. Let       C    ∞    (      R    ) be the space of all smooth functions       R    →      R  ; this is naturally a R-vector space, albeit of infinite dimension. Consider the operator   D  :      C    ∞    (      R    )  →      C    ∞    (      R    ) defined by  D  f  =      ∑          k      =      0              n            a    k        f          (      k      )      where       a    k    :      R    →      R   are some smooth functions and       f          (      k      )       is the k-th derivative of f. D is easily seen to be a linear operator, and a differential equation of the form  D  f  =  gis precisely a linear ODE of order n. That is to say, the word ‘linear’ refers to the linearity of D as an operator! In this light, solving a linear ODE consists of two steps:1. Finding the kernel of D, i.e. solving the homogeneous linear ODE   D  f  =  0, and2. Finding a ‘particular integral’       f    0   such that   D      f    0    =  g.Step 2This is exactly the same as solving a system of (possibly inhomogeneous) linear equations in ordinary linear algebra!Now, one might be tempted to find an analogue of Gaussian elimination to work with linear ODEs, but the fact that       C    ∞    (      R    ) has infinite dimension and no natural basis tends to screw things up a little...

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