Question

Verify if \[x+-\frac{3}{2}(x+y-2)+\ln|x+y-2|=c\] is a solution to \[(x+y) dx + (3x+3y-4)dy = 0\]

Answer 1

Or you could start with the latter equation and see if it ends up with the former equation if it's too hard. I substituted y with
\[t=a_1x+b_1y => y = \frac{t-a_1x}{b_1}\]
and it follows that\[dy=\frac{dt-a_1dx}{b_1}\]

Answer 2

diff equation can be done by sub \(z=x+y\)

Answer 3

\[\frac{dy}{dx}=\frac{x+y}{4-3(x+y)}\]

Answer 4

Well I have to solve it in a specific way. It's a homogeneous differential equation where M(x,y) and N(x,y) are linear but not homogeneous. The equations in M(x,y) and N(x,y) actually represent equations of lines, which in this case are parallel.

Answer 5

sorry i was out

so u want to make it homogeneous with mapping \((x,y)\) to \((X,Y)\) so that the diff equation will be homogeneous in tems of \(X\) and \(Y\)

Answer 6

u can show that the sub \(X=x-x_0\) and \(Y=y-y_0\) will change the nonhomogeneous

diff equation\[\frac{dy}{dx}=\frac{ax+by+c}{ex+fy+g}\]to homogeneous diff equation\[\frac{dY}{dX}=\frac{aX+bY}{eX+fY}\]where \((x_0,y_0)\) is solution of system\[ax+by+c=0\]\[ex+fy+g=0\]