Is there a closed form for any function f(x,y) satisfying: \frac{df}{dx}+\frac{df}{dy}=xy

ajedrezlaproa6j 2022-01-18 Answered
Is there a closed form for any function f(x,y) satisfying:
dfdx+dfdy=xy
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Anzante2m
Answered 2022-01-19 Author has 34 answers
Forgive for using Mathematica, but the answer are:
f(x,y)=16(3x2yx3)+F(yx)
with F - arbitrary continuously differentiable function
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Charles Benedict
Answered 2022-01-20 Author has 32 answers
alenahelenash
Answered 2022-01-24 Author has 343 answers
Is solved by f(x,y)=xy2/2y3/6.One way to get this: Let fx=a(x,y) so that fy=xya(x,y). Then use commutativity of mixed partials to get a handle on a; we'd have ay(x,y)=yax(x,y). Then, why not set ax=0? we get ay=y and so a(x,y)=y22 works fine. Since we can solve exact differential equations, we should be able to solve this given that (fx)y=(fy)x, indeed we can, with the above
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