Answer to Question #316648 in Differential Equations for Tobias Felix

Question

Coefficients linear in two variables

(3x − y + 2) dx + (9x − 3y + 1)dy = 0

Answer 1

$\left( 3x-y+2 \right) dx+\left( 9x-3y+1 \right) dy=0\\u=3x-y+2\\x=\frac{u+y-2}{3}\\dx=\frac{1}{3}\left( du+dy \right) \\u\cdot \frac{1}{3}\left( du+dy \right) +\left( 3u-5 \right) dy=0\\\left( 10u-5 \right) dy+udu=0\\y=-\int{\frac{udu}{10u-5}}=-\left( \frac{1}{10}\int{du}+\frac{1}{2}\int{\frac{1}{10u-5}du} \right) =\\=-\frac{1}{10}u+\frac{1}{20}\ln \left| u-\frac{1}{2} \right|+C\\y=-\frac{1}{10}\left( 3x-y+2 \right) +\frac{1}{20}\ln \left| 3x-y+\frac{3}{2} \right|+C$