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

ajedrezlaproa6j

ajedrezlaproa6j

Answered question

2022-01-18

Is there a closed form for any function f(x,y) satisfying:
dfdx+dfdy=xy

Answer & Explanation

Anzante2m

Anzante2m

Beginner2022-01-19Added 34 answers

Forgive for using Mathematica, but the answer are:
f(x,y)=16(3x2yx3)+F(yx)
with F - arbitrary continuously differentiable function
Charles Benedict

Charles Benedict

Beginner2022-01-20Added 32 answers

Solution: f(x,y)=x3+3x2y+6c1(yx)6 (1)
alenahelenash

alenahelenash

Expert2022-01-24Added 556 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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