Can someone explain why this special transformation work?
suppose we have a differential equation in standard form.
$$M(x,y)dx+N(x,y)dy=0$$
If the differential equation is in this special form
$$(a_1x+b_1y+c_1)dx+(a_2x+b_2y+c_2)dy=0$$
then,
Let (h,k) be the solution to the differential equation such that
$$a_1h+b_1k+c_1=0$$ $$a_2h+b_2k+c_2=0$$
The transformation is then
$$x=X+h$$
$$y=Y+k$$
I can't understand why this works! Can someone provide both an intuitive explanation and proof if there is. I just explore the world of differential equation.
It is obvious that the above differential equation is not in homogeneous form, Bernoulli's, separable variable or exact form!
the basic idea is you get rid of the terms $c_{1}$ and $c_{2}$ by solving for $a_{1}h+b_{1}k+c_{1} = 0$ and $a_{2}h+b_{2}k+c_{2}=0$, and now that you don't have any constants the equation turns into a simple homogeneous diff equation.. just remember to subsitue back $X = x - h$ and $Y = y-k$