Reduction to separable variables

 

Special Case

$\frac{\mathrm{d}y}{\mathrm{d}x} = f(Ax + By + C)$     $(B \neq 0)$

 

# If $B=0$, then $\frac{\mathrm{d}y}{\mathrm{d}x} = f(Ax + By + C)$ is separable.

 

 

 

Solution

$u = Ax + By + C$  $\Rightarrow$  $\frac{\mathrm{d}u}{\mathrm{d}x} = A + B\frac{\mathrm{d}y}{\mathrm{d}x}$

 

$\frac{\mathrm{d}y}{\mathrm{d}x} = f(Ax + By + C)$

     $\Rightarrow$  $\frac{1}{B} \frac{\mathrm{d}u}{\mathrm{d}x} -\frac{A}{B}= f(u)$

     $\Rightarrow$  $\frac{\mathrm{d}u}{\mathrm{d}x} = Bf(u) + A$

It is separable.