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I am trying to solve the following PDE $$\bigg(p \cdot \frac{\partial}{\partial p}+q \cdot \frac{\partial}{\partial q} +3\bigg)F_{1}(p,q)=F_{0}(p,q)$$ where $F_{1}$ and $F_{0}$ are two functions of $\vec{p}$, $\vec{q}$ and the operator on the left of the PDE involves the scalar products between a vector and the corresponding gradient. The right part of the equation is known to be $$F_{0}(p,q)=\frac{\pi^3}{p^2q^2(p-q)^2}$$ where the squares denotes the scalar product of a vector with itself. Given these conditions, what can be said about a generic solution of $F_{1}$?

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It is a quasilinear equation we can apply the Lagrange method. Since $pF_p+qF_q=F_0-3F$, we have $\frac{dp}{p}=\frac{dq}{q}=\frac{dF}{F_0-3F}$. We can find its first integrals by solving these equalities. From the first two equaality (it is an ODE for $p$ and $q$) we immediateyl find that $p-q=c_1$, or $p=q+c_1$. Now taking the second and third equality and considering $p=q+c_1$, we get another ODE (for $q$ and $F$) $(\frac{\pi^3}{q^2(q+c_1)^2c_1^2}-3F)dq-qdF=0$ with an integrating factor $q.$ After multiplying the eqution by this factor we get an exact differential equation: Solving this you'll get a second integral with arbitrary constant $c_2$. I think you can proceed from here. Your solution will be an implicit one either in the form $c_1=g(c_2)$ or $G(c_1,c_2)=0$ for arbitrary continuously differentiable function $g$ or $G$. Here $F=F_1$

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