Answer to Question #308115 in Differential Equations for Papi Chulo

Question #308115

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)=

1
Expert's answer
2022-03-11T03:32:32-0500

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


The equation can be separated as, dy8y+5=(4x+5)dx\dfrac{dy}{8y+5} = (4x+5)dx


Integrating, we get


18log(8y+5)=2x2+5x+c\dfrac{1}{8}\log (8y+5) = 2x^2 + 5x + c


Therefore,

F(x,y)=G(x)+H(y)=2x2+5xlog(8y+5)8F(x,y)=G(x)+H(y)= 2x^2 + 5x - \dfrac{\log (8y+5)}{8}.


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