$-A\frac{\partial^{2}Q}{\partial x^{2}}+B\frac{\partial^{3}P}{\partial x^{3}}+CQ-C\frac{\partial P}{\partial x}+D\frac{\partial^{2}Q}{\partial y^{2}}=0\tag{{1}}$ $-B\frac{\partial^{3}Q}{\partial x^{3}}+E\frac{\partial^{4}P}{\partial x^{4}}-C\frac{\partial Q}{\partial x}+C\frac{\partial^{2}P}{\partial x^{2}}-F\frac{\partial^{2}P}{\partial y^{2}}=0\tag{{2}}$
Differentiate $(1)$ with respect to $x$ :$-A\frac{\partial^{3}Q}{\partial x^{3}}+B\frac{\partial^{4}P}{\partial x^{4}}+C\frac{\partial Q}{\partial x}-C\frac{\partial^{2}P}{\partial x^{2}}+D\frac{\partial^{3}Q}{\partial x\partial y^{2}}=0\tag{{3}}$
Add $(2)$ and $(3)$:$-\left(A+B\right)\frac{\partial^{3}Q}{\partial x^{3}}+\left(E+B\right)\frac{\partial^{4}P}{\partial x^{4}}+D\frac{\partial^{3}Q}{\partial x\partial y^{2}}-F\frac{\partial^{2}P}{\partial y^{2}}=0\tag{4}$ Differentiate $(2)$ twice with respect to $y$ :$-B\frac{\partial^{5}Q}{\partial x^{3}\partial y^{2}}+E\frac{\partial^{6}P}{\partial x^{4}\partial y^{2}}-C\frac{\partial^{3}Q}{\partial x\partial y^{2}}+C\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}-F\frac{\partial^{4}P}{\partial y^{4}}=0\tag{{5}}$
Differentiate $(4)$ twice with respect to $x$ $-\left(A+B\right)\frac{\partial^{5}Q}{\partial x^{5}}+\left(E+B\right)\frac{\partial^{6}P}{\partial x^{6}}+D\frac{\partial^{5}Q}{\partial x^{3}\partial y^{2}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}=0\tag{6}$
Differentiate $(2)$ twice with respect to $x$ :$-B\frac{\partial^{5}Q}{\partial x^{5}}+E\frac{\partial^{6}P}{\partial x^{6}}-C\frac{\partial^{3}Q}{\partial x^{3}}+C\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}=0\tag{{7}}$
From $(7)$:
$\frac{\partial^{5}Q}{\partial x^{5}}=\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{6}}-C\frac{\partial^{3}Q}{\partial x^{3}}+C\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}\right)\tag{{8}}$
Substitute into $(6)$: $-\left(A+B\right)\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{6}}-C\frac{\partial^{3}Q}{\partial x^{3}}+C\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}\right)+\left(E+B\right)\frac{\partial^{6}P}{\partial x^{6}}+D\frac{\partial^{5}Q}{\partial x^{3}\partial y^{2}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}=0 \tag{9}$
From $(5)$ $\frac{\partial^{5}Q}{\partial x^{3}\partial y^{2}}=\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{4}\partial y^{2}}-C\frac{\partial^{3}Q}{\partial x\partial y^{2}}+C\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}-F\frac{\partial^{4}P}{\partial y^{4}}\right)\tag{10}$
Substitute into $(9)$: $-\left(A+B\right)\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{6}}-C\frac{\partial^{3}Q}{\partial x^{3}}+C\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}\right)+\left(E+B\right)\frac{\partial^{6}P}{\partial x^{6}}+D\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{4}\partial y^{2}}-C\frac{\partial^{3}Q}{\partial x\partial y^{2}}+C\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}-F\frac{\partial^{4}P}{\partial y^{4}}\right)-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}=0\tag{11}$
From $(4)$: $\frac{\partial^{3}Q}{\partial x\partial y^{2}}=-\frac{1}{D}\left(-\left(A+B\right)\frac{\partial^{3}Q}{\partial x^{3}}+\left(E+B\right)\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{2}P}{\partial y^{2}}\right)\tag{12}$
Substitute into $(11)$: $-\left(A+B\right)\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{6}}-C\frac{\partial^{3}Q}{\partial x^{3}}+C\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}\right)+\left(E+B\right)\frac{\partial^{6}P}{\partial x^{6}}+D\frac{1}{B}\left(E\frac{\partial^{6}P}{\partial x^{4}\partial y^{2}}+C\frac{1}{D}\left(-\left(A+B\right)\frac{\partial^{3}Q}{\partial x^{3}}+\left(E+B\right)\frac{\partial^{4}P}{\partial x^{4}}-F\frac{\partial^{2}P}{\partial y^{2}}\right)+C\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}-F\frac{\partial^{4}P}{\partial y^{4}}\right)-F\frac{\partial^{4}P}{\partial x^{2}\partial y^{2}}=0\tag{13}$
Differentiate $(1)$ with respect to $x$ three times:$-A\frac{\partial^{5}Q}{\partial x^{5}}+B\frac{\partial^{6}P}{\partial x^{6}}+C\frac{\partial^{3}Q}{\partial x^{3}}-C\frac{\partial^{4}P}{\partial x^{4}}+D\frac{\partial^{5}Q}{\partial y^{5}}=0\tag{14}$
$(13)$ gives an expression for $\frac{\partial^{3}Q}{\partial x^{3}}$ . substitution into $(8)$ gives equation for $\frac{\partial^{5}Q}{\partial x^{5}}$ . Substituting the results in $(14)$ gives an equation in terms of P and its derivatives only.