Class 12

Math

Calculus

Differential Equations

The differential equations, find a particular solution satisfying the given condition: $dydx +2ytanx=sinx;y=0$when $x=3π $

$dxdy +Py=Q(x)$, we get,$P=2tanxandQ(x)=sinx$

So, Integrating factor $(I.F)=e_{∫2tanxdx}$

$I.F.=e_{2lnsecx}=e_{lnsec_{2}x}=sec_{2}x$

We know, solution of differential equation,

$y(I.F.)=∫Q(I.F.)dx$

$∴$Our solution will be,

$ysec_{2}x=∫sinx(sec_{2}x)dx$

$⇒ysec_{2}x=∫cos_{2}xsinx dx$

$⇒ysec_{2}x=∫tanxsecxdx$

$⇒ysec_{2}x=secx+c$

$⇒y=cosx+ccos_{2}x$

At $x=3π ,y=0$

$⇒0=21 +4c $

$⇒−2=c$

So, our solution will be,

$⇒y=cosx−2cos_{2}x$

$⇒y=cosx(1−2cosx)$, which is the required solution.