How to find the number of arbitrary constants in solution of a given differential equation ?
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$\begingroup$In general, the number of arbitrary constants of an ordinary differential equation (ODE) is given by the order of the highest derivative.
e.g. $$y\color{red}'=f(x)$$ has $\color{red}{\text{one}}$ constant of integration, $$y\color{red}{''}+y'=f(x)$$ has $\color{red}{\text{two}}$, $$y\color{red}{'''}+y''=f(x)$$ has $\color{red}{\text{three}},$ etc..
To generalise, suppose we've an ODE of the form $$a_ny^{(\color{green}{n})}+a_{n-1}y^{^{(n-1)}}+\cdots+a_2y''+a_1y'+a_0y=f(x), $$ where $a_i$ is a function of $x$.
Then the number of arbitrary constants in the general solution to this equation is $\color{green}{n}.$
Notation: $y^{(n)}=\frac{d^ny}{dx^n}$
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