D'Alembert's Differential Equation

equation

y(x)x

2

′

y

(x)

+

2

′

y

(x)

parameter of general solution

0

{x(p),y(p)}

a

2

(p-1)

+

2

p

2

-

3

p

3

2

(p-1)

,

2

p

a

2

(p-1)

+

2

p

2

-

3

p

3

2

(p-1)

+

2

p



D'Alembert's (or Lagrange’s) differential equation has the form

y=xf(p)+g(p)

, (1)

where

p=y'

. Differentiating the equation with respect to

x

, we get

p=f(p)+[xf'(p)+g'(p)]dp/dx

. (2)

This equation is linear with respect to

x(p)

:

(p-f(p))/(dp/dx)-xf'(p)=g'(p)

. (2')

From this, we get the solution

x=Ch(p)+j(p)

. (3)

Here

h(p)

is a solution of the corresponding homogeneous equation of (2’) and

j(p)

is a particular solution of (2’). Equations (1) and (3) determine the solution parametrically. Eliminating the parameter

p

(if possible), we get the general solution in the form

F(x,y,C)=0

.