Question

(1+tanx)dy=2siny dx solve differential equation?

Answer 1

`y=2arc tan{c(sinx+cosx)e^x},` is the GS.

Explanation:

Rewriting the given Diff. Eqn., as,

`dy/siny=2/(1+tanx)dx,` we find that, it is Separable Variable type.

`:. cscydy=(2cosx)/(sinx+cosx)dx.`

The General Solution (GS) is obtained by integrating term-wise.

`:. intcsc y dy=int(2cosx)/(sinx+cosx)dx+lnc.`

`:. ln(tan(y/2))=int{(sinx+cosx)+(cosx-sinx)}/(sinx+cosx)dx+lnc.`

`=int{(sinx+cosx)/(sinx+cosx)+(cosx-sinx)/(sinx+cosx)}dx+lnc,`

`=int1dx+int1/tdt,`

`where, sinx+cosx=t rArr (cosx-sinx)dx=dt.`

`:. ln(tan(y/2))=x+lnt+lnc, i.e., x+ln(sinx+cosx)+lnc.`

`:. ln(tan(y/2))-ln(sinx+cosx)=x+lnt, or,`

`ln{(tan(y/2))/(sinx+cosx)}=lne^x+lnc=ln(ce^x),`

`:. tan(y/2)/(sinx+cosx)=ce^x.`

`:. tan(y/2)=c(sinx+cosx)e^x.`

`:. y/2=arc tan{c(sinx+cosx)e^x}.`

`rArr y=2arc tan{c(sinx+cosx)e^x},` is the desired GS.

Enjoy Maths.!