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Why is this a linear differential equation? 
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#1
Jan1912, 10:14 PM

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Why is y'2xy=x a linear differential equation? I thought it would be nonlinear due to the 2xy...?



#2
Jan1912, 10:30 PM

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#3
Jan2012, 11:42 AM

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#4
Jan2012, 12:33 PM

P: 150

Why is this a linear differential equation?
Another way of looking at it is to consider the operator [itex]L=\frac{d}{dx}2x[/itex], so that the differential equation becomes [itex]Ly=x[/itex]. Then we say the differential equation is linear if that operator [itex]L[/itex] is linear, i.e. [tex]L(f+g)=L(f)+L(g)[/tex] and [tex]L(cf)=cL(f)[/tex] for all (suitably smooth) functions [itex]f[/itex] and [itex]g[/itex] and constants [itex]c[/itex] (where addition of functions and multiplication of a function by a constant are defined pointwise as usual). This is equivalent to what Ray Vickson just said: if you consider the equation [itex]Ly=0[/itex], then (as a result of the linearity of [itex]L[/itex]) any linear combination of solutions is also a solution.



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