Question #284297

Solve the equation in exact differentials:

(y-3x2)dx+(x-4y)dy=0


1
Expert's answer
2022-01-09T10:35:29-0500

We shall solve the equation

(y3x2)dx+(x4y)dy(y-3x^{2})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)=y3x2M(x,y)=y-3x^{2} and

N(x,y)=x4yN(x,y)=x-4y

We know that for exactness,

M(x,y)y\frac{\partial M(x,y)}{\partial y} must be equal to N(x,y)x\frac{\partial N(x,y)}{\partial x}


Now,

M(x,y)y=1\frac{\partial M(x,y)}{\partial y}=1 and N(x,y)x=1\frac{\partial N(x,y)}{\partial x}=1

Showing that the differential equation is EXACT.


We shall then solve the equation,

Fx(x,y)=y3x2(1)Fy(x,y)=x4y(2)F_{x}(x,y)=y-3x^{2}------(1)\\F_{y}(x,y)=x-4y------(2)

Integrating equ(1) w.r.t.w.r.t. xx

F(x,y)=xyx3+ϕ(y)()F(x,y)=xy-x^{3}+\phi(y)---(*)

Differentiating equ()equ(*) w.r.t.w.r.t. yy

Fy(x,y)=x+ϕ(y)()F_{y}(x,y)=x+\phi^{\prime}(y)----(**)

Equating equ(2)equ(2) and equ()equ(**)

x4y=x+ϕ(y)    ϕ(y)=4yx-4y=x+\phi^{\prime}(y)\\\implies\phi^{\prime}(y)=-4y\\

Integrating both sides w.r.t yy

ϕ(y)=2y2\phi(y)=-2y^{2}


Now, from the equation

F(x,y)=xyx3+ϕ(y)F(x,y)=xy-x^{3}+\phi(y)


Substituting the value of ϕ(y)=2y2\phi(y)=-2y^{2}


We have

F(x,y)=xyx32y2F(x,y)=xy-x^{3}-2y^{2}

Which is our solution to the differential equation.








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