Solve the differential equation of the following exact differential equation. Show complete solution.
(3x2y - 6x)dx + (x3 + 2y)dy = 0
Let M(x,y)=3x2y−6x ∫(3x2y−6x)dx=x3y−3x2+h(y)Differentiating and comparing to N(x,y)=x3+2y, we have thath′(y)=2y ⟹ h(y)=y2Hence the solution to the given exact equation isx3y−3x2+y2=C\displaystyle \text{Let $M(x,y) = 3x^2y-6x$ }\\ \int (3x^2y-6x)dx = x^3y-3x^2+h(y)\\ \text{Differentiating and comparing to $N(x,y) = x^3+2y$, we have that}\\ h'(y) =2y \implies h(y) = y^2\\ \text{Hence the solution to the given exact equation is}\\ x^3y-3x^2+y^2=CLet M(x,y)=3x2y−6x ∫(3x2y−6x)dx=x3y−3x2+h(y)Differentiating and comparing to N(x,y)=x3+2y, we have thath′(y)=2y⟹h(y)=y2Hence the solution to the given exact equation isx3y−3x2+y2=C
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