# Why is this a linear differential equation?

by SMA_01
Tags: differential, equation, linear
 P: 218 Why is y'-2xy=x a linear differential equation? I thought it would be nonlinear due to the 2xy...?
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 Quote by SMA_01 Why is y'-2xy=x a linear differential equation? I thought it would be nonlinear due to the 2xy...?
It is linear *in y*; it would be a nonlinear DE if it contained things like y^2, exp(y), 1/(1+y), etc. Another way to see it is: if y1 and y2 are two solutions and a, b are constants, then the linear combination a*y1 + b*y2 is also a solution. That would generally fail for a nonlinear DE.

RGV
 P: 150 Why is this a linear differential equation? Another way of looking at it is to consider the operator $L=\frac{d}{dx}-2x$, so that the differential equation becomes $Ly=x$. Then we say the differential equation is linear if that operator $L$ is linear, i.e. $$L(f+g)=L(f)+L(g)$$ and $$L(cf)=cL(f)$$ for all (suitably smooth) functions $f$ and $g$ and constants $c$ (where addition of functions and multiplication of a function by a constant are defined point-wise as usual). This is equivalent to what Ray Vickson just said: if you consider the equation $Ly=0$, then (as a result of the linearity of $L$) any linear combination of solutions is also a solution.