Solvable form of differential equation
An inexact differential equation is a differential equation of the form (see also: inexact differential )
M
(
x
,
y
)
d
x
+
N
(
x
,
y
)
d
y
=
0
,
where
∂
M
∂
y
≠
∂
N
∂
x
.
{\displaystyle M(x,y)\,dx+N(x,y)\,dy=0,{\text{ where )){\frac {\partial M}{\partial y))\neq {\frac {\partial N}{\partial x)).}
The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739.[1]
Solution method
In order to solve the equation, we need to transform it into an exact differential equation . In order to do that, we need to find an integrating factor
μ
{\displaystyle \mu }
to multiply the equation by. We'll start with the equation itself.
M
d
x
+
N
d
y
=
0
{\displaystyle M\,dx+N\,dy=0}
, so we get
μ
M
d
x
+
μ
N
d
y
=
0
{\displaystyle \mu M\,dx+\mu N\,dy=0}
. We will require
μ
{\displaystyle \mu }
to satisfy
∂
μ
M
∂
y
=
∂
μ
N
∂
x
{\textstyle {\frac {\partial \mu M}{\partial y))={\frac {\partial \mu N}{\partial x))}
. We get
∂
μ
∂
y
M
+
∂
M
∂
y
μ
=
∂
μ
∂
x
N
+
∂
N
∂
x
μ
.
{\displaystyle {\frac {\partial \mu }{\partial y))M+{\frac {\partial M}{\partial y))\mu ={\frac {\partial \mu }{\partial x))N+{\frac {\partial N}{\partial x))\mu .}
After simplifying we get
M
μ
y
−
N
μ
x
+
(
M
y
−
N
x
)
μ
=
0.
{\displaystyle M\mu _{y}-N\mu _{x}+(M_{y}-N_{x})\mu =0.}
Since this is a partial differential equation , it is mostly extremely hard to solve, however in some cases we will get either
μ
(
x
,
y
)
=
μ
(
x
)
{\displaystyle \mu (x,y)=\mu (x)}
or
μ
(
x
,
y
)
=
μ
(
y
)
{\displaystyle \mu (x,y)=\mu (y)}
, in which case we only need to find
μ
{\displaystyle \mu }
with a first-order linear differential equation or a separable differential equation , and as such either
μ
(
y
)
=
e
−
∫
M
y
−
N
x
M
d
y
{\displaystyle \mu (y)=e^{-\int ((\frac {M_{y}-N_{x)){M))\,dy))}
or
μ
(
x
)
=
e
∫
M
y
−
N
x
N
d
x
.
{\displaystyle \mu (x)=e^{\int ((\frac {M_{y}-N_{x)){N))\,dx)).}