Answer:
![f(x,y)=2x^2+4y^2+2xy=C_1\\\\Where\\\\y(x)=\frac{1}{4} (-x\pm \sqrt{-7x^2+C_1} )](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D2x%5E2%2B4y%5E2%2B2xy%3DC_1%5C%5C%5C%5CWhere%5C%5C%5C%5Cy%28x%29%3D%5Cfrac%7B1%7D%7B4%7D%20%28-x%5Cpm%20%5Csqrt%7B-7x%5E2%2BC_1%7D%20%29)
Step-by-step explanation:
Let:
![M(x,y)=4x+2y\\\\and\\\\N(x,y)=2x+8y](https://tex.z-dn.net/?f=M%28x%2Cy%29%3D4x%2B2y%5C%5C%5C%5Cand%5C%5C%5C%5CN%28x%2Cy%29%3D2x%2B8y)
This is and exact equation, because:
![\frac{\partial M(x,y)}{\partial y} =2=\frac{\partial N}{\partial x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20M%28x%2Cy%29%7D%7B%5Cpartial%20y%7D%20%3D2%3D%5Cfrac%7B%5Cpartial%20N%7D%7B%5Cpartial%20x%7D)
So, define f(x,y) such that:
![\frac{\partial f(x,y)}{\partial x} =M(x,y)\\\\and\\\\\frac{\partial f(x,y)}{\partial y} =N(x,y)](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20f%28x%2Cy%29%7D%7B%5Cpartial%20x%7D%20%3DM%28x%2Cy%29%5C%5C%5C%5Cand%5C%5C%5C%5C%5Cfrac%7B%5Cpartial%20f%28x%2Cy%29%7D%7B%5Cpartial%20y%7D%20%3DN%28x%2Cy%29)
The solution will be given by:
![f(x,y)=C_1](https://tex.z-dn.net/?f=f%28x%2Cy%29%3DC_1)
Where C1 is an arbitrary constant
Integrate
with respect to x in order to find f(x,y):
![f(x,y)=\int\ {4x+2y} \, dx =2x^2+2xy+g(y)](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D%5Cint%5C%20%7B4x%2B2y%7D%20%5C%2C%20dx%20%3D2x%5E2%2B2xy%2Bg%28y%29)
Where g(y) is an arbitrary function of y.
Differentiate f(x,y) with respect to y in order to find g(y):
![\frac{\partial f(x,y)}{\partial y} =2x+\frac{d g(y)}{dy}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20f%28x%2Cy%29%7D%7B%5Cpartial%20y%7D%20%3D2x%2B%5Cfrac%7Bd%20g%28y%29%7D%7Bdy%7D)
Substitute into ![\frac{\partial f(x,y)}{\partial y} =N(x,y)](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20f%28x%2Cy%29%7D%7B%5Cpartial%20y%7D%20%3DN%28x%2Cy%29)
![2x+\frac{dg(y)}{dy} =2x+8y\\\\Solve\hspace{3}for\hspace{3}\frac{dg(y)}{dy}\\\\\frac{dg(y)}{dy}=8y](https://tex.z-dn.net/?f=2x%2B%5Cfrac%7Bdg%28y%29%7D%7Bdy%7D%20%3D2x%2B8y%5C%5C%5C%5CSolve%5Chspace%7B3%7Dfor%5Chspace%7B3%7D%5Cfrac%7Bdg%28y%29%7D%7Bdy%7D%5C%5C%5C%5C%5Cfrac%7Bdg%28y%29%7D%7Bdy%7D%3D8y)
Integrate
with respect to y:
![g(y)=\int\ {8y} \, dy =4y^2](https://tex.z-dn.net/?f=g%28y%29%3D%5Cint%5C%20%7B8y%7D%20%5C%2C%20dy%20%3D4y%5E2)
Substitute g(y) into f(x,y):
![f(x,y)=2x^2+4y^2+2xy](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D2x%5E2%2B4y%5E2%2B2xy)
The solution is f(x,y)=C1
![f(x,y)=2x^2+4y^2+2xy=C_1](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D2x%5E2%2B4y%5E2%2B2xy%3DC_1)
Solving y using quadratic formula:
![y(x)=\frac{1}{4} (-x\pm \sqrt{-7x^2+C_1} )](https://tex.z-dn.net/?f=y%28x%29%3D%5Cfrac%7B1%7D%7B4%7D%20%28-x%5Cpm%20%5Csqrt%7B-7x%5E2%2BC_1%7D%20%29)