Question

Find the complete integral of the equation (∂z/∂x_1)(∂z/∂x_2)(∂z/∂x_3)=z³x_1x_2x_3?

Answer 1

One solution is:

`z(bbx) = " const " * \ e^(1/2( C_1 x_1^2 + C_2 x_2^2 + C_3 x_3^2 ))`

Explanation:

Symmetry suggests that separation of variables should result in a solution:

So let `z(bbx) = X_1(x_1) \ X_2(x_2) \ X_3(x_3)`

`:. partial_(x_1) \z = X_1^' \ X_2 \ X_3`

....with corresponding results for `partial_(x_2) \z` and `partial_(x_3) \z`.

So the PDE now reads:

• #X_1^' X_2^' X_3^' (X_1 X_2 X_3)^2 = (X_1 X_2 X_3)^3 \ x_1 x_2 x_3#

• `X_1^' \ X_2^' \ X_3^' = x_1 X_1 \ x_2 X_2 \ x_3 X_3`

• `(X_1^')/(X_1 \ x_1) = ( x_2 X_2)/(X_2^') * (x_3 X_3)/(X_3^'`

This is the same statement as:

• `f(x_1) = 1/(g(x_2)) * 1/(h(x_3)) qquad forall x_i`

These must be constant functions, or at very least they can be, so make an assumption or draw a conclusion:

`{(f_1 = C_1),(g_2 = C_2),(h_3 = C_3):}`

• `" where " C_1 = 1/(C_2 C_3)`

It follows that:

`(d X_1)/(X_1 ) = C_1\ x_1 \ d x_1`

Integrating:

`ln X_1 = C_1\ x_1^2/2 + D_1`

• `X_1 = d_1e^( C_1\ x_1^2/2 )`

Pattern-matching:

• `{(X_2 = d_2e^( C_2\ x_2^2/2 )),(X_3 = d_3e^( C_3\ x_3^2/2 )):}`

`color(blue)( z(bbx) = d \ e^(1/2( C_1 x_1^2 + C_2 x_2^2 + C_3 x_3^2 ))) qquad d = color(green)(d_1 d_2 d_3)`

Validating this possible solution:

• `partial_(x_1) z = C_1 x_1 \ d_1 \ e^(1/2( C_1 x_1^2 + C_2 x_2^2 + C_3 x_3^2 ))`

• `partial_(x_2) z = C_2 x_2 \ d_2 \ e^(1/2( C_1 x_1^2 + C_2 x_2^2 + C_3 x_3^2 ))`

• `partial_(x_3) z = C_3 x_3\ d_3 \ e^(1/2( C_1 x_1^2 + C_2 x_2^2 + C_3 x_3^2 ))`

`partial_(x_1) z \ partial_(x_2) \ partial_(x_3) z = \ C_1 C_2 C_3 \ x_ 1 x_2x_3\ d^3 \ e^(3/2( c_1 x_1^2 + c_2 x_2^2 + c_3 x_3^2 ))`

And because, by separating the variables, a specific conclusion was drawn:

• `C_1 = 1/(C_2 C_3)`

`= x_ 1 x_2x_3\ underbrace( d^3 \ e^(3/2( c_1 x_1^2 + c_2 x_2^2 + c_3 x_3^2 )))_(= z^3)`

`= z^3 \ x_1x_2x_3`

So, at least one solution exists. Whether or not it is the exact solution depends upon the boundary values.