Maple can easily solve the B4 problem of the Putnam Mathematical Competition 2019  link


B4.  Let F be the set of functions f(x,y) that are twice continuously differentiable for x≥1, y≥1 and that satisfy the following two equations:
    x*(diff(f(x, y), x))+y*(diff(f(x, y), y)) = x*y*ln(x*y)

x^2*(diff(f(x, y), x, x))+y^2*(diff(f(x, y), y, y)) = x*y


For each f2F, let


"m(f) = min[s>=1]  (f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s))"


Determine m(f), and show that it is independent of the choice of f.


# Solution

x*diff(f(x,y),x)+y*diff(f(x,y),y) = x*y*ln(x*y),
x^2*diff(f(x,y),x,x)+y^2*diff(f(x,y),y,y) = x*y

{f(x, y) = (1/2)*(x*y+2*_C1)*ln(x*y)-(1/2)*x*y-2*_C1*ln(x)+_C2}


f:=unapply(rhs(%[]), x,y);

proc (x, y) options operator, arrow; (1/2)*(y*x+2*_C1)*ln(y*x)-(1/2)*y*x-2*_C1*ln(x)+_C2 end proc


h := f(s+1, s+1) - f(s+1, s) - f(s, s+1) + f(s, s);



minimize(h, s=1..infinity);



answer = simplify(%);

answer = 2*ln(2)-1/2




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