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:

$\frac{df}{dx}+\frac{df}{dy}=xy$

You can still ask an expert for help

Anzante2m

Answered 2022-01-19
Author has **34** answers

Forgive for using Mathematica, but the answer are:

$f(x,y)=\frac{1}{6}(3{x}^{2}y-{x}^{3})+F(y-x)$

with F - arbitrary continuously differentiable function

with F - arbitrary continuously differentiable function

Charles Benedict

Answered 2022-01-20
Author has **32** answers

Solution: $f(x,y)=\frac{-{x}^{3}+3{x}^{2}y+6{c}_{1}(y-x)}{6}$ (1)

alenahelenash

Answered 2022-01-24
Author has **343** answers

Is solved by $f(x,y)=x{y}^{2}/2-{y}^{3}/6$ .One way to get this: Let ${f}_{x}=a(x,y)$ so that ${f}_{y}=xy-a(x,y)$ . Then use commutativity of mixed partials to get a handle on a; we'd have ${a}_{y}(x,y)=y-{a}_{x}(x,y)$ . Then, why not set ${a}_{x}=0$ ? we get ${a}_{y}=y$ and so $a(x,y)=\frac{{y}^{2}}{2}$ works fine. Since we can solve exact differential equations, we should be able to solve this given that $({f}_{x}{)}_{y}=({f}_{y}{)}_{x}$ , indeed we can, with the above

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