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The differential equation

dy/dx=(6x+9)/(12y^2+16y+6)

has an implicit general solution of the form F(x,y)=K.
In fact, because the differential equation is separable, we can define the solution curve implicitly by a function in the form
F(x,y)=G(x)+H(y)=K.
Find such a solution and then give the related functions requested.
F(x,y)=G(x)+H(y)=

Respuesta :

LammettHash

As the question points out, the equation is indeed separable:

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{6x+9}{12y^2+16y+6}[/tex]

[tex]\implies2(6y^2+8y+3)\,\mathrm dy=3(2x+3)\,\mathrm dx[/tex]

Integrate both sides to get

[tex]2(2y^3+4y^2+3y)=3(x^2+3x)+C[/tex]

[tex]\implies F(x,y)=\underbrace{(-3x^2-9x)}_{G(x)}+\underbrace{4y^3+8y^2+6y}_{H(y)}=C[/tex]

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