In the book it is written that the order of a differential equation is the same as the number of arbitrary constants present in the original equation.
So, if there are 'n' arbitrary constants in the equation, we have to differentiate it 'n' times to remove those constants and form a differential equation.
There is a question which asks to form a differential equation free from arbitrary constants of $ax + by + c = 0$
This is my attempt to it :-
$$ax + by + c = 0\\=> ax + by = -c\\\text{differentiating both sides w.r.t. x, we get}\\a + by' = 0\\\text{again differentiating both sides w.r.t. x, we get}\\by'' = 0\\=> y'' = 0$$
On differentiating it only 2 times, all three arbitrary constants are removed. Do I have to differentiate it again to form $y''' = 0$ ?
Best Answer
Not necessarily.
We can rewrite it as
$ax + by +c = 0$
Dividing throughout by b,
$\frac{a}{b}x + y + \frac{c}{b} = 0$
Let $A = \frac{a}{b}$ and $B = \frac{c}{b}$
Thus,
$Ax + y + B= 0$
or $- y = Ax +B$
Now it has only 2 arbitrary constants.
So, we need to differentiate only twice.