Class 12

Math

Calculus

Differential Equations

Find the general solution of the differential equations:$(x+3y_{2})dydx =y(y>0)$

$(x+3y_{2})dxdy =y$

$⇒dydx =yx+3y_{2} $

$⇒dydx −yx =3y$

Comparing it with $dydx +Px=Q$

$P=−y1 andQ=3y$

Integrating factor, $I.F.=e_{∫Pdy}$

$I.F.=e_{∫−y1dy}=e_{−lny}=y_{−lne}=y−1=y1 $

Now, general solution will be,

$x(I.F.)=∫(I.F.)Qdy$

$⇒yx =∫(y1 )(3y)dy$

$⇒yx =3y+c$

$⇒x=3y_{2}+cy$