Question

General solution of (1+x)day/dx-xy=1-x

Answer 1

Let x + y = v

⇒ 1+dydx=dvdx⇒ 1+dydx=dvdx

or dydx =dvdx−1dydx =dvdx−1

So, (x+y+1)dydx=1(x+y+1)dydx=1 can be re-written as:

(v+1)(dvdx−1)=1(v+1)(dvdx−1)=1

(v+1)dvdx−v−1=1(v+1)dvdx−v−1=1

dvdx =v+2v+1dvdx =v+2v+1 ; which is variable separable form

dx =v+1v+2dvdx =v+1v+2dv

dx =(1−1v+2)dvdx =(1−1v+2)dv

Integrating both the sides:

x =v−ln(v+2)+lnCx =v−ln(v+2)+lnC

v−x=ln(v+2)−lnCv−x=ln(v+2)−lnC

v−x =ln(v+2C)v−x =ln(v+2C)

v+2C =ev−xv+2C =ev−x

v+2=Cev−x