We shall solve the equation
(y−3x2)dx+(x−4y)dy
In exact differentials.
First of all, we shall show that the differential equation is indeed EXACT.
From the equation, we have
M(x,y)=y−3x2 and
N(x,y)=x−4y
We know that for exactness,
∂y∂M(x,y) must be equal to ∂x∂N(x,y)
Now,
∂y∂M(x,y)=1 and ∂x∂N(x,y)=1
Showing that the differential equation is EXACT.
We shall then solve the equation,
Fx(x,y)=y−3x2−−−−−−(1)Fy(x,y)=x−4y−−−−−−(2)
Integrating equ(1) w.r.t. x
F(x,y)=xy−x3+ϕ(y)−−−(∗)
Differentiating equ(∗) w.r.t. y
Fy(x,y)=x+ϕ′(y)−−−−(∗∗)
Equating equ(2) and equ(∗∗)
x−4y=x+ϕ′(y)⟹ϕ′(y)=−4y
Integrating both sides w.r.t y
ϕ(y)=−2y2
Now, from the equation
F(x,y)=xy−x3+ϕ(y)
Substituting the value of ϕ(y)=−2y2
We have
F(x,y)=xy−x3−2y2
Which is our solution to the differential equation.
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