From Wikipedia, the free encyclopedia
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]
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}}.}
Classification
Operations Attributes of variables Relation to processes
Solutions
Existence/uniqueness Solution topics Solution methods
Examples Mathematicians