If $a\frac {dy}{dx} + by = c$ has constant coeffcients, does that means that $a=b=c$? I am trying to identify if a differential equation has constant coefficients.
Let $A = a\dfrac {dy}{dx} + by = c$
The $A$ has constant coefficients only if $a=b=c$ correct?
 A: The general form of an ODE is
$$F(x,y,y',y'',...,y^{[n]})=g(x)$$
We call it homogeneous if $g(x)=0$. (Well, this isn't technically precise, but I hope you get the idea.) We call it linear if it is of the form
$$p_0(x)y+p_1(x)y'+...+p_n(x)y^{[n]}=g(x)$$
We call it constant coefficient if it is of the form
$$c_0q_0(y)+c_1q_1(y')+...+c_nq_n(y^{[n]})=g(x)$$
Where $c_0,...,c_n$ are constants but are not necessarily equal.
A: Let us look to some examples:
Homogeneous equations with constant coefficients:

*

*$y'+y=0$

*$2y'+y=0$

*$y'+3y=0$

*$2y'+3y=0$
Nonhomogeneous equations with constant coefficients:

*

*$y'+y=1$

*$2y'+y=1+x$

*$y'+3y=x^2$

*$2y'+3y=e^x$
Homogeneous equations with variable coefficients:

*

*$xy'+y=0$

*$y'+x^2y=0$

*$e^xy'+y=0$

*$xy'+e^xy=0$
Nonhomogeneous equations with variable coefficients:

*

*$xy'+y=1$

*$y'+x^2y=1+x$

*$e^xy'+y=x^2$

*$xy'+e^xy=e^x$
In general, a first-order linear differential equation has constant coefficients if it has the form
$$a\frac{dy}{dx}+by=c$$
with $a,b,c\in\mathbb R$ and $a\neq 0$.
