Solve x^2 dx - z^2 dy - xy dz = 0
Solve \[ x^2\,dx - z^2\,dy - xy\,dz = 0. \]
Explanation of Method
Test the Frobenius integrability condition; when it fails, attempt to reduce the characteristic system to a chain of solvable ODEs. Here that reduction runs into a genuinely nonlinear second-order equation with no elementary closed-form solution.
Step 1 · Test integrability
With $P=x^2$, $Q=-z^2$, $R=-xy$: \[ P(R_y-Q_z)+Q(P_z-R_x)+R(Q_x-P_y) = x^2(2z-x)-yz^2 \neq 0. \] No integral surface exists, so we look for the two-parameter family of integral curves of the characteristic system \[ \frac{dx}{x^2}=\frac{dy}{-z^2}=\frac{dz}{-xy}. \]
Step 2 · Reduce to a system of ODEs in $x$
Treating $x$ as the independent variable, \[ \frac{dy}{dx} = -\frac{z^2}{x^2}, \qquad \frac{dz}{dx} = -\frac{y}{x}. \] From the second equation $y=-xz'$, so $y'=-z'-xz''$. Substituting into the first equation gives a single second-order equation for $z(x)$: \[ xz''+z' = \frac{z^2}{x^2}. \]
Step 3 · Assess closed-form solvability
This is a genuinely nonlinear second-order ODE (no linear substitution reduces it to a standard solvable type — it is not Bernoulli, Euler–Cauchy, or exact in $z,z',z''$), and a systematic search for algebraic first integrals $F(x,y,z)$ with $x^2F_x-z^2F_y-xyF_z\equiv0$ (polynomial and simple rational ansätze) finds none. No elementary closed-form solution exists for this equation.
Result
Conclusion: