(3x^2+2xy)dx+(2x^2y^3-x^2)dy
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Integrate M with respect to x, writing the constant of integration as a function of y (or integrate N with respect to y, writing the constant as a function of x);
Differentiate the result with respect to y (x) and equate to N (M); and
Integrate with respect to y(x) to determine that constant of integration from step 1.
This DE is exact, because ∂M∂y=∂N∂x=4x+2y, so:
F(x,y)=∫(3x2+4xy+y2)dx=x3+2x2y+xy2+g(y) … or F(x,y)=∫(2x2+2xy+9)dy=2x2y+xy2+9y+h(x)
∂F∂y=2x2+2xy+g′(y)=2x2+2xy+9, so g′(y)=9 … or ∂F∂x=4xy+y2+h′(x)=3x2+4xy+y2, so h′(x)=3x2
g(y)=∫9dy=9y … or h(x)=∫3x2dx=x3
Whichever of the above approaches you follow, you get F(x,y)=x3+2x2y+xy2+9y. (Note that we could have taken a shortcut past steps 2 and 3 by comparing the two answers we got in step 1.) The solutions of this DE are level curves of this function. Because this is quadratic in y, one could solve for an explicit solution; here are some solution curves: