Question

Solve $(x+2y)(dx-dy)=dx+dy$.

Answer 1

4 \ln|3x+6y-1| + 6(y-x) = K

Answer 2

The differential equation is transformed into the form $\frac{dy}{dx} = f(ax+by+c)$. A substitution $v=x+2y$ is used to convert it into a separable differential equation in $v$ and $x$. This separable equation is solved by integration. Finally, the substitution is reversed to obtain the general solution in terms of $x$ and $y$. The case where the denominator in the separated variable form is zero leads to a singular solution.