Your equations are:

q = qq[x, y]; p = pp[x, y];

r1 = -a D[q, {x, 2}] +b D[p, {x, 3}] +c D[q, {x, 0}] -c D[p, {x, 1}] + d D[q, {y , 2}] == 0;

s1 = -b D[q, {x, 3}] +e D[p, {x, 4}] -c D[q, {x, 1}] +c D[p, {x, 2}] - f D[p, {y , 2}] == 0;


then declare:

rule = Flatten[{#[x, y] -&gt; #, Derivative[n_, m_][#][x_, y_] -&gt; # x^n y^m} &amp; /@ {pp, qq}];
invRule = (x_ -&gt; y_) :&gt; y Derivative[Sequence @@ x][pp];


Transform  

t = Total[CoefficientRules[Eliminate[{r1, s1} /. rule, qq] /. (x_ == y_) -&gt; (x - y), {x, y}] 
      /. pp -&gt; 1 /. invRule] == 0;

t // StandardForm // TeXForm


$c f \text{pp}^{(0,2)}+d f \text{pp}^{(0,4)}+(-c d-a f) \text{pp}^{(2,2)}+(a c-c e) \text{pp}^{(4,0)}-d e
   \text{pp}^{(4,2)}+\left(-b^2+a e\right) \text{pp}^{(6,0)}=0$

How can I eliminate a function in a system of PDEs, Maple-style?
What do you call a former criminal who has served their sentence and been released from prison?

I thought of “convicted criminal”, but that might imply the person is still a criminal and/or serving a sentence.


Does “convicted criminal” fit this definition?
Are there other--better--words that are more accurate?

