# Complete vs General Integral of first order PDE

In Russian *"integral"* is a synonym of a solution of differential equation. *"general integral"* means general solution, *"complete"* probably means sum of particular solution and general solution (called the complementary solution)

iPDEs

In the general theory of partial differential equations and specifically for *First-Order Partial Differential Equations* one defines the *general solution*(Landau's general integral) and the *complete integral* as follows:

For a two-dimensional first order partial differential equation $$f(x,y,z,z_x,z_y)=0. \tag{1}$$

Complete Integral: A two parameter family of implicit solutions of the form (2) of (1) is called a complete integral of the partial differential equation. $$\phi(x,y,z,a,b)=0. \tag{2}$$General solution:A function of the form (3), where $u(x,y,z)$ and $v(x,y,z)$ are functions of $x,y,z$ and $\Phi$ is an arbitrary smooth function, $\Phi$ is called a (implicit or explicit) general solution of (1), if $z,z_x,z_y$ as determined by the relation (3) satisfy (1) $$\Phi(u,v)=0\tag{3}.$$

*If we have a complete integral (2) of (1), we can derive a general solution (3), we would show this later in the post but first, let's see how to derive the PDE (1) from the complete integral (2).

If we have a complete integral (2), we can obtain $d\phi/dx$ and $d\phi/dy$ : $$\phi_x+z_x\phi_z=0. \tag{4}$$ $$\phi_y+z_y\phi_z=0. \tag{5}$$

With (2),(4),(5) we can obtain an expression of the form (1) that is free from the parameters $a$ and $b$. If (1) is obtained exactly from (2),(4),(5) then $\phi$ is a solution of the PDE (1).

Now, to derive a general solution (3) from a complete integral (2), we can impose $b=W(a)$ in the complete solution (2), obtaining $\Phi(x,y,z,a,W(a))$, and impose the condition $d\Phi/da=0$, $$\frac{d\Phi}{da}=\Phi_a(x,y,z,a,W(a))+W'(a)\Phi_W(x,y,z,a,W(a))=0. \tag{6}$$

With (6) we can write $a=A(x,y,z)$ as a function of $x,y,z$. So the general solution derived from (2) can be written as $$\Phi\Big(x,y,z,A(x,y,z),W(A(x,y,z))\Big) = 0. \tag{7}$$

We can see that (7) in fact matches our definition of general solution.Now we will prove that (7) is a solution of (1), again $d\Phi/dx$ and $d\Phi/dy$

$$\Phi_x+z_x\Phi_z+ \Phi_A A_x+\Phi_W W'(A) A_x =0. \tag{8}$$ $$\Phi_y+z_y\Phi_z+ \Phi_A A_y+\Phi_W W'(A) A_y =0. \tag{9}$$

Now applying the condition (6) Equations (8), and (9) yield:

$$\Phi_x+z_x\Phi_z =0. \tag{8}$$ $$\Phi_y+z_y\Phi_z =0. \tag{9}$$

Now the systems of equations (2),(4),(5) yield the same derived expression (1) as (7),(8),(9), now $\Phi\Big(x,y,z,A(x,y,z),W(A(x,y,z))\Big)$ is a general solution of (1) and we can see that we obtain a different solution for every function $W$.

We can see that this solution is free of the parameters $a$ and $b$, when we choose a particular function $W$ we obtain a *particular solution* for the PDE.

Landau's generalizes this result in his footnote, however he does it for an easier equation, not a general First-order PDE (1). The steps he does are the same as we did for a general two dimensional First-order PDE.

The notion "complete integral" here refers to solutions of specific (1st order) PDEs that depend on the maximal number of constants of motion. If you want a concrete example I can refer you to Equation (10) of this paper, or even better to Ref. [10] in that paper.

A "general solution", by contrast, need not depend explicitly on constants of motion, but usually contains some free (integration) function. As an example for a solution not depending explicitly on constants of motion see the "enveloping solution" in Eq. (11) of the paper above.

[I never encountered these notions anywhere, except when solving Hamilton-Jacobi equations - this seems to be also the context to which Landau refers.]