Question #292778

solve the equation in exact differentials

(2-9xy2)xdx+(4y2-6x3)ydy=0


1
Expert's answer
2022-02-08T07:25:52-0500

Let us solve the equation in exact differentials (29xy2)xdx+(4y26x3)ydy=0.(2-9xy^2)xdx+(4y^2-6x^3)ydy=0.


It follows that there exists the function u=u(x,y)u=u(x,y) such that


ux=(29xy2)x=2x9x2y2, uy=(4y26x3)y=4y36x3y.\frac{\partial u}{\partial x}=(2-9xy^2)x=2x-9x^2y^2, \ \frac{\partial u}{\partial y}=(4y^2-6x^3)y=4y^3-6x^3y.


Therefore, u=x23x3y2+c(y),u=x^2-3x^3y^2+c(y), and hence


uy=6x3y+c(y)=4y36x3y.\frac{\partial u}{\partial y}=-6x^3y+c'(y)=4y^3-6x^3y.


It follows that c(y)=4y3,c'(y)=4y^3, and thus c(y)=y4+C.c(y)=y^4+C.


We conclude that the general solution of the differential equation is


x23x3y2+y4=C.x^2-3x^3y^2+y^4=C.


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