If $a\frac {dy}{dx} + by = c$ has constant coeffcients, does that means that $a=b=c$?

Question

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?

Answer 1

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.

Answer 2

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 $a\neq 0$.