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?
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1$\begingroup$ It's not a linear differential equation so you can obtain almost everything, there is not rule when it's nonlinear. $\endgroup$– LelouchAug 28, 2022 at 19:35
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2 Answers
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You have two solution families, $y=x+c$ and $y=2x+c$, each with one parameter.
The statement about the number of integration constants applies only to explicit ODE, where the highest derivatives are isolated or can be uniquely determined. This does not apply to an DE where the highest derivative appears in a quadratic polynomial.
answered Aug 29, 2022 at 12:10
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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'
