How to solve the following differential equation $\frac{dy}{dx} =\frac{x+2y+8}{2x+y+7}$

Question

How to solve the following differential equation $\displaystyle\frac{dy}{dx} =\frac{x+2y+8}{2x+y+7}$.

Answer 1

$$\frac{dy}{dx} =\frac{x+2y+8}{2x+y+7}$$

Hint: Introduce new variables $X$, $Y$ such that $$x=X+a,\quad\text{and} \quad y=Y+b$$ by choosing appropriate constants $a$, $b$ in order to express your diffeq as an equivalent expression without the constants $8$ and $7$ terms on the right side of your give equation.

In terms of the new variables, your equation will then have the form $$\frac{dY}{dX}=\frac{X+2Y}{ 2X+Y} =\frac{1+2{Y\over X}}{ 2+1{Y\over X}}\ .$$