My Professor said that a first order linear differential equation is of the following form :
$\frac{dy}{dx}+Py=Q$, where $P$ and $Q$ are functions of $x$.
I am unable to understand why is this true.
I understand that $P$ should be a function of $x$, otherwise there would be a term of higher powers of $y$. But I am unable to understand why $Q$ should be a function of $x$.
Why can't we take $Q$ as $y$? If we take $Q=y$, then it would still be first order linear differential equation. I am unable to understand why we should take $Q$ as a function of $x$, why can't we take $Q$ as $y$.
Can someone please help me out?
I would be grateful if someone could clear my doubt.
It is a Definition where we use Placeholders $P$ & $Q$
Yes , $Q$ too can have $y$ , then we have to move the terms around & rename the Placeholders
$dy/dx+x^2y=x^3+x+1 \tag{1}$
Here $P=x^2$ & $Q=x^3+x+1$ : It will match the Definition & we can claim that we have linear first order ODE
$dy/dx+x^2y=x^3y+x+1 \tag{2.1}$
Here $P=x^2$ & $Q=x^3y+x+1$ : It will not match the Definition
Hence we move the terms around to get :
$dy/dx+x^2y-x^3y=x+1 \tag{2.2}$
It is that Exact Same Equation !!
Here $P=x^2-x^3$ & $Q=x+1$ : It will match the Definition & we can claim that we have a linear first order ODE
No matter what $y$ linear terms we have , we move it all together & use the Placeholder $P$ to represent that
We will move together all other terms involving $x$ & constants & use the Placeholder $Q$ to represent that
Definition than says that when we have that form , we can claim that it is a linear first order ODE