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I arrived at the following problem from performing a separation of variables on a system of two PDEs in two unknowns.

Let $\{Y_1(y),Y_2(y)\}$ be two given non-zero functions of $y$, $\{F(x),G(x)\}$ be two unknown functions we seek to find, and $\{L_1,L_2,\ldots,L_8\}$ be eight linear operators (polynomial differential operators in $x$ with constant coefficients). I have the following system of ODEs:

$Y_1\big\{L_1F+L_2G\big\}+Y_2\big\{L_3F+L_4G\big\}=0$

$Y_1\big\{L_5F+L_6G\big\}+Y_2\big\{L_7F+L_8G\big\}=0$

A sufficient condition for a solution is that all bracketed terms vanish simultaneously. This gives us a system of four equations with only two unknown functions.

I was thinking the following method would work. First, add the two above equations to get

$Y_1\big\{(L_1+L_5)F+(L_2+L_6)G\big\}+Y_2\big\{(L_3+L_7)F+(L_4+L_8)G\big\}=0$

and then demand that the two bracketed terms vanish simultaneously. This then gives us a system of two equations with two unknowns, which we know how to solve.

Is this kosher?

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    Are $Y_1$, $Y_2$ *linear* functions? Then you could include such constant multiplications into the "operators". All this is a bit vague, though... – 2017-01-30
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    No, $Y_1$ and $Y_2$ are nonlinear functions. – 2017-01-30
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    Then you cannot add the equations and include everything into the parentheses. And also it is not true that $Y_1(A)+Y_2(B)=0$ implies $A=B=0$. Now the ODEs are nonlinear and you have no guarantee that solutions exist. – 2017-01-31

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