Homogeneous type of first-order differential equations Homogeneous differential equation

a first-order ordinary differential equation in form:

m
(
x
,
y
)

d
x
+
n
(
x
,
y
)

d
y
=
0


{\displaystyle m(x,y)\,dx+n(x,y)\,dy=0}

is homogeneous type if both functions m(x, y) , n(x, y) homogeneous functions of same degree n. is, multiplying each variable parameter  



λ


{\displaystyle \lambda }

, find

m
(
λ
x
,
λ
y
)
=

λ

n


m
(
x
,
y
)



{\displaystyle m(\lambda x,\lambda y)=\lambda ^{n}m(x,y)\,}

    ,    



n
(
λ
x
,
λ
y
)
=

λ

n


n
(
x
,
y
)

.


{\displaystyle n(\lambda x,\lambda y)=\lambda ^{n}n(x,y)\,.}

thus,

m
(
λ
x
,
λ
y
)


n
(
λ
x
,
λ
y
)



=



m
(
x
,
y
)


n
(
x
,
y
)




.


{\displaystyle {\frac {m(\lambda x,\lambda y)}{n(\lambda x,\lambda y)}}={\frac {m(x,y)}{n(x,y)}}\,.}



solution method

in quotient  






m
(
t
x
,
t
y
)


n
(
t
x
,
t
y
)



=



m
(
x
,
y
)


n
(
x
,
y
)





{\displaystyle {\frac {m(tx,ty)}{n(tx,ty)}}={\frac {m(x,y)}{n(x,y)}}}

, can let  



t
=
1

/

x


{\displaystyle t=1/x}

  simplify quotient function



f


{\displaystyle f}

of single variable



y

/

x


{\displaystyle y/x}

:

m
(
x
,
y
)


n
(
x
,
y
)



=



m
(
t
x
,
t
y
)


n
(
t
x
,
t
y
)



=



m
(
1
,
y

/

x
)


n
(
1
,
y

/

x
)



=
f
(
y

/

x
)

.


{\displaystyle {\frac {m(x,y)}{n(x,y)}}={\frac {m(tx,ty)}{n(tx,ty)}}={\frac {m(1,y/x)}{n(1,y/x)}}=f(y/x)\,.}

introduce change of variables



y
=
u
x


{\displaystyle y=ux}

; differentiate using product rule:

d
(
u
x
)


d
x



=
x



d
u


d
x



+
u



d
x


d
x



=
x



d
u


d
x



+
u
,


{\displaystyle {\frac {d(ux)}{dx}}=x{\frac {du}{dx}}+u{\frac {dx}{dx}}=x{\frac {du}{dx}}+u,}

thus transforming original differential equation separable form

x



d
u


d
x



=
f
(
u
)
−
u

;


{\displaystyle x{\frac {du}{dx}}=f(u)-u\,;}

this form can integrated directly (see ordinary differential equation).

the equations in discussion not used formulary solutions; shown demonstrate method of solution.

special case

a first order differential equation of form (a, b, c, e, f, g constants)

(
a
x
+
b
y
+
c
)
d
x
+
(
e
x
+
f
y
+
g
)
d
y
=
0



{\displaystyle (ax+by+c)dx+(ex+fy+g)dy=0\,}

where af ≠ can transformed homogeneous type linear transformation of both variables (



α


{\displaystyle \alpha }

,



β


{\displaystyle \beta }

constants):

t
=
x
+
α
;




z
=
y
+
β

.


{\displaystyle t=x+\alpha ;\,\,\,\,z=y+\beta \,.}






^ ince 1956, p. 18