Question

The differential equation dy/dx = 25+20x+40y+32xy

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 of F(x,y)=G(x)+H(y)=K

Find such a solution and then give the related functions requested for

F(x,y)=G(x)+H(y)=

Answer 1

The given equation can be rewritten as, $\dfrac{dy}{dx} = (4x+5)(8y+5)$.

The equation can be separated as, $\dfrac{dy}{8y+5} = (4x+5)dx$

Integrating, we get

$\dfrac{1}{8}\log (8y+5) = 2x^2 + 5x + c$

Therefore,

$F(x,y)=G(x)+H(y)= 2x^2 + 5x - \dfrac{\log (8y+5)}{8}$.