D'Alembert's Differential Equation
D'Alembert's Differential Equation
D'Alembert's (or Lagrange’s) differential equation has the form
y=xf(p)+g(p)
where . Differentiating the equation with respect to , we get
p=y'
x
p=f(p)+[xf'(p)+g'(p)]dp/dx
This equation is linear with respect to :. (2')
x(p)
(p-f(p))/(dp/dx)-xf'(p)=g'(p)
From this, we get the solution
x=Ch(p)+j(p)
Here is a solution of the corresponding homogeneous equation of (2’) and is a particular solution of (2’). Equations (1) and (3) determine the solution parametrically. Eliminating the parameter (if possible), we get the general solution in the form .
h(p)
j(p)
p
F(x,y,C)=0