# Relation between number of arbitrary constants and degree of differential equation

Say we have a differential equation:
$$(\frac{dy}{dx})^2 - 3\frac{dy}{dx} + 2 = 0$$
Upon solving, we get:
$$\frac{dy}{dx}=1,2$$
$$y=x+c_1, 2x+c_2$$
So the solution of the DE, according to me, should be
$$(y-x-c_1)(y-2x-c_2)=0$$
But we have been taught that the number of arbitrary constants only depends on the order of the DE. I was told that if we use 2 different constants, then it'll be impossible to get rid of them after differentiating only once (since the order is 1) and that I should the same constant $$c$$ in both brackets. But the curve we get with 2 different constants seems to satisfy the DE.
Can someone tell me where I'm making a mistake?

• It's not a linear differential equation so you can obtain almost everything, there is not rule when it's nonlinear. Aug 28, 2022 at 19:35

You have two solution families, $$y=x+c$$ and $$y=2x+c$$, each with one parameter.
Write your De as $$(y'-1)*(y'-2)=0$$ and you see that you have 2 independent De . With independent solutions , you can make more DE of this kind, by multiplying are terms with y'