Binomial differential equation

In mathematics, the binomial differential equation is an ordinary differential equation of the form

(

y

′

)

m

=

f

(

x

,

y

)

,

{\displaystyle \left(y'\right)^{m}=f(x,y),}

where

m

{\displaystyle m}

is a natural number and

f

(

x

,

y

)

{\displaystyle f(x,y)}

is a polynomial that is analytic in both variables.

Solution

Let

P

(

x

,

y

)

=

(

x

+

y

)

k

{\displaystyle P(x,y)=(x+y)^{k}}

be a polynomial of two variables of order

k

{\displaystyle k}

, where

k

{\displaystyle k}

is a natural number. By the binomial formula,

P

(

x

,

y

)

=

∑

j

=

0

k

(

k

j

)

x

j

y

k

−

j

{\displaystyle P(x,y)=\sum \limits _{j=0}^{k}{{\binom {k}{j}}x^{j}y^{k-j}}}

.

The binomial differential equation becomes

(

y

′

)

m

=

(

x

+

y

)

k

{\textstyle (y')^{m}=(x+y)^{k}}

. Substituting

v

=

x

+

y

{\displaystyle v=x+y}

and its derivative

v

′

=

1

+

y

′

{\displaystyle v'=1+y'}

gives

(

v

′

−

1

)

m

=

v

k

{\textstyle (v'-1)^{m}=v^{k}}

, which can be written

d

v

d

x

=

1

+

v

k

m

{\textstyle {\tfrac {dv}{dx}}=1+v^{\tfrac {k}{m}}}

, which is a separable ordinary differential equation. Solving gives

d

v

d

x

=

1

+

v

k

m

⇒

d

v

1

+

v

k

m

=

d

x

⇒

∫

d

v

1

+

v

k

m

=

x

+

C

{\displaystyle {\begin{array}{lrl}&{\frac {dv}{dx}}&=1+v^{\tfrac {k}{m}}\\\Rightarrow &{\frac {dv}{1+v^{\tfrac {k}{m}}}}&=dx\\\Rightarrow &\int {\frac {dv}{1+v^{\tfrac {k}{m}}}}&=x+C\end{array}}}

Special cases

If

m

=

k

{\displaystyle m=k}

, this gives the differential equation

v

′

−

1

=

v

{\displaystyle v'-1=v}

and the solution is

y

(

x

)

=

C

e

x

−

x

−

1

{\displaystyle y\left(x\right)=Ce^{x}-x-1}

, where

C

{\displaystyle C}

is a constant.

If

m

|

k

{\displaystyle m|k}

(that is,

m

{\displaystyle m}

is a divisor of

k

{\displaystyle k}

), then the solution has the form

∫

d

v

1

+

v

n

=

x

+

C

{\textstyle \int {\frac {dv}{1+v^{n}}}=x+C}

.