Class 12

Math

Calculus

Differential Equations

Show that the differential equation $xcos(xy )dxdy =ycos(xy )+x$is homogeneous and solve it.

Then, $y=vx⇒dxdy =v+xdxdv $

So, given equation becomes,

$xcosv(dxdy )=vxcosv+x$

$⇒dxdy =xcosvvxcosv+x $

$⇒v+xdxdv =cosvvcosv+1 $

$⇒v+xdxdv =v+secv$

$⇒xdxdv =secv$

$⇒cosdv=xdx $

Integrating both sides,

$⇒sinv=ln(x)+c$

$⇒sin(xy )=ln(x)+c$, which is the required equation.