Question

The differential equation dydx=15+30x+18y+36xy 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)=

Answer 1

`(x + 2x^2)-(1)/(18)\ln |15+18y|=C'`

Explanation:

Given differential equation,

`(dy)/(dx)=15+30x+18y + 36xy`

`(dy)/(dx)=15(1+2x)+18y(1+2x)`

`(dy)/(dx)=(15+18y)(1+2x)`

`(dy)/(15+18y)=(1+2x)dx------(1)`

Let 15 + 18y = t

18dy = dt

`\implies dy=(dt)/(18)`

From equation (1),

`(1)/(18) (1)/(t)dt = (1+2x)dx`

Integrating both sides,

`(1)/(18)\ln |t| = x + 2x^2 + C`

`(1)/(18)\ln |15+18y| = x + 2x^2 + C`

`\implies (x + 2x^2)-(1)/(18)\ln |15+18y|=C'` ( where C' = -C = constant)

Which is the required equation,

Where,

`G(x) = x+2x^2\text{ and }H(y) = -(1)/(18)\ln |15+18y|`