Constant coefficients in a differential equation

silly_penguin88

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How can i tell when a differential equation has a constant coefficient?
Ex.1) y''+3y'-2y=sinx yes constant coefficient. ok makes sense.
Ex.2) y'-2y=xy No constant coefficient. Why not? isn't -2y a constant coefficient?
 
How can i tell when a differential equation has a constant coefficient?
Ex.1) y''+3y'-2y=sinx yes constant coefficient. ok makes sense.
Ex.2) y'-2y=xy No constant coefficient. Why not? isn't -2y a constant coefficient?
Yes, the -2 in front of the y is constant. However, a constant coefficient differential equation means that if we collect ALL terms involving each derivative and the function itself into a single coefficient each and, if those coefficients are constant, then it is a differential equation with constant coefficients.

For Ex. 1: Collecting all terms for each we have the coefficients for
y'' is 1
y' is 3
y is -2
Those are all constants and, as you say, ok makes sense.
For Ex. 2: Since the equation is equivalent to
y'-2y - xy = 0
and, collecting all terms, we have
y' - (2 + x) y = 0
we have the coefficients for
y' is 1
y is -(2 + x)
So the coefficient for y is not constant and we do not have a differential equation with constant coefficients.
 
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