Question Number 82245 by jagoll last updated on 19/Feb/20
![find the solution x sin ((y/x)) dy = [y sin ((y/x)) −x] dx](Q82245.png)
$${find}\:{the}\:{solution}\: \\ $$$${x}\:\mathrm{sin}\:\left(\frac{{y}}{{x}}\right)\:{dy}\:=\:\left[{y}\:\mathrm{sin}\:\left(\frac{{y}}{{x}}\right)\:−{x}\right]\:{dx} \\ $$
Commented by john santu last updated on 19/Feb/20
![let v = (y/x) ⇒ y = vx (dy/dx) = v + (dv/dx) ⇒ dy =v dx+ dv (x sin v)(v dx + dv) = [vx sin v−x]dx xv sin v dx + xsin v dv = xv sin v dx−x dx x sin v dv = −x dx sin v dv = −dx ⇒∫ sin v dv = −x+c −cos v = −x +c cos ((y/x)) = x + C](Q82246.png)
$${let}\:{v}\:=\:\frac{{y}}{{x}}\:\Rightarrow\:{y}\:=\:{vx} \\ $$$$\frac{{dy}}{{dx}}\:=\:{v}\:+\:\frac{{dv}}{{dx}}\:\Rightarrow\:{dy}\:={v}\:{dx}+\:{dv}\: \\ $$$$\left({x}\:\mathrm{sin}\:{v}\right)\left({v}\:{dx}\:+\:{dv}\right)\:=\:\left[{vx}\:\mathrm{sin}\:{v}−{x}\right]{dx} \\ $$$${xv}\:\mathrm{sin}\:{v}\:{dx}\:+\:{x}\mathrm{sin}\:{v}\:{dv}\:=\:{xv}\:\mathrm{sin}\:{v}\:{dx}−{x}\:{dx} \\ $$$${x}\:\mathrm{sin}\:{v}\:{dv}\:=\:−{x}\:{dx} \\ $$$$\mathrm{sin}\:{v}\:{dv}\:=\:−{dx}\:\Rightarrow\int\:\mathrm{sin}\:{v}\:{dv}\:=\:−{x}+{c} \\ $$$$−\mathrm{cos}\:{v}\:=\:−{x}\:+{c}\: \\ $$$$\mathrm{cos}\:\left(\frac{{y}}{{x}}\right)\:=\:{x}\:+\:{C}\: \\ $$
Commented by jagoll last updated on 19/Feb/20
$${thank}\:{sir} \\ $$