The solution of the differential equation e^ -x (y+1) d y+ ( ² x- 2 x ) y(d x)=0 subjected to the condition that y=1 , when x=0 is

Step-by-step solution

Given, differential equation is aligned & e^ -x (y+1) d y+ ( ² x- 2 x ) y d x=0 & ( y+1 y ) d y+e^ x ( ² x- 2 x ) d x=0 & (1+ 1 y ) d y+ (e^ x ² x-e^ x 2 x ) d x=0 & d(y+ y)+d (e^ x ² x )=0 aligned On integrating above equation, we get aligned &y+ y+e^ x ² x=c & At y(0)=1 & 1+0+e⁰ 1=c c=2 aligned Hence, required solution is y+ y+e^ x ² x=2