Separation of Variables | Equations of Order One

Given the differential equation

$M(x, y)\,dx + N(x, y)\,dy = 0 \,\, \to \,\,$ Equation (1)

where $\,M\,$ and $\,N\,$ may be functions of both $\,x\,$ and $\,y\,$. If the above equation can be transformed into the form

$f(x)\,dx + f(y)\,dy = 0\,\, \to \,\,$ Equation (2)

where $\,f(x)\,$ is a function of $\,x\,$ alone and $\,f(y)\,$ is a function of $\,y\,$ alone, equation (1) is called variables separable.

To find the general solution of equation (1), simply equate the integral of equation (2) to a constant $\,c\,$. Thus, the general solution is

$\displaystyle \int f(x)\,dx + \int f(y)\,dy = c$


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