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

Answer:

$(x + 2x^2)-\frac{1}{18}\ln |15+18y|=C'$

Step-by-step explanation:

Given differential equation,

$\frac{dy}{dx}=15+30x+18y + 36xy$

$\frac{dy}{dx}=15(1+2x)+18y(1+2x)$

$\frac{dy}{dx}=(15+18y)(1+2x)$

$\frac{dy}{15+18y}=(1+2x)dx------(1)$

Let 15 + 18y = t

18dy = dt

$\implies dy=\frac{dt}{18}$

From equation (1),

$\frac{1}{18} \frac{1}{t}dt = (1+2x)dx$

Integrating both sides,

$\frac{1}{18}\ln |t| = x + 2x^2 + C$

$\frac{1}{18}\ln |15+18y| = x + 2x^2 + C$

$\implies (x + 2x^2)-\frac{1}{18}\ln |15+18y|=C'$   ( where C' = -C = constant)

Which is the required equation,

Where,

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