First Order ODE in form y' = F ((a x + b y + c) over (d x + e y + f)) where a e = b d

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Theorem

Formulation 1

The first order ODE:

$(1): \quad \dfrac {\d y} {\d x} = F \left({\dfrac {a x + b y + c} {d x + e y + f} }\right)$

such that:

$a e = b d$


can be solved by substituting:

$z = a x + b y$


to obtain:

$\dfrac {\d z} {\d x} = b F \left({\dfrac {a z + a c} {d z + f} }\right) + a$

which can be solved by the technique of Separation of Variables.


Formulation 2

The first order ODE:

$(1): \quad \dfrac {\d y} {\d x} = \map F {\dfrac {a x + b y + c} {d x + e y + f} }$

such that:

$a e = b d$


can be solved by substituting:

$z = d x + e y$


to obtain:

$\dfrac {\d z} {\d x} = e \map F {\dfrac {b z + e c} {e z + e f} } + d$

which can be solved by the technique of Separation of Variables.


Examples

$\dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x + y - 6}$

The first order ODE:

$(1): \quad \dfrac {\d y} {\d x} = \dfrac {x + y + 4} {x + y - 6}$

has the general solution:

$y - x = 5 \, \map \ln {x + y - 1} + C$


Sources