Solving the equation xzx yzy=z

To solve the equation xzx yzy = z, we can start by separating the variables and integrating. We will also use the chain rule to simplify the integrals. Let's begin by rewriting the equation in terms of differentials:

xzx dx + yzy dy = dz

Now, let's integrate both sides of the equation. We will integrate with respect to x on the left side and with respect to z on the right side:

∫xzx dx + ∫yzy dy = ∫dz

Integrating with respect to x:

To integrate xzx dx, we use the chain rule. Let's set u = x^2, so du = 2x dx:

∫xzx dx = ∫(x^2)(x dx) = ∫u du/2 = u^2/4 = (x^2)^2/4 = x^4/4

Integrating with respect to y:

To integrate yzy dy, we use the chain rule. Let's set v = y^2, so dv = 2y dy:

∫yzy dy = ∫(y^2)(y dy) = ∫v dv/2 = v^2/4 = (y^2)^2/4 = y^4/4

Integrating with respect to z:

The integral of dz is simply z:

∫dz = z

Putting it all together:

Now let's substitute the integrals back into the original equation:

x^4/4 + y^4/4 = z

This is the solution to the equation xzx yzy = z.

The curve that satisfies the associated characteristics equations

To find the curve that satisfies the associated characteristics equations, we need to solve the following system of equations:

dx/dt = xz
dy/dt = yz
dz/dt = z

Using the chain rule and the given equation x^2 y^2 = a^2, we can rewrite the system of equations as:

dx/x = z dt
dy/y = z dt
dz/z = dt

Integrating each equation separately:

∫dx/x = ∫z dt
∫dy/y = ∫z dt
∫dz/z = ∫dt

ln|x| = ∫z dt + C1
ln|y| = ∫z dt + C2
ln|z| = ∫dt + C3

Where C1, C2, and C3 are constants of integration.

Exponentiating each equation:

x = e^(∫z dt + C1)
y = e^(∫z dt + C2)
z = e^(∫dt + C3)

Simplifying the exponents:

x = e^(C1) * e^(∫z dt)
y = e^(C2) * e^(∫z dt)
z = e^(C3) * e^t

Finally, combining the equations and substituting btan^(-1)(y/x) for z:

x^2 y^2 = a^2

(e^(C1) * e^(∫z dt))^2 * (e^(C2) * e^(∫z