COMEDK2015MathematicsDifferential Equations
The general solution of the differential equation ( d y d x )+y g^ (x)=g(x) g^ i (x) , where g(x) is a given function of x is
Options
- Ag(x)+ (1+y+g(x))=c
- Bg(x)+ (1+y-g(x))=c
- Cg(x)- (1+y-g(x))=c
- Dg(x)- (1-y+g(x))=c
Correct answer
B. g(x)+ (1+y-g(x))=c
Step-by-step solution
Given, differential equation is d y d x +y g^ (x)=g(x) g^ (x) This is in the form of linear differential equation. So, IF =e^ g^ (x) d x =e^ g(x) Required solution is y e^ g(x) = g(x) g^ (x) e^ g(x) d x+c^ Put g(x)=t g^ (x) d x=d t y e^ g(x) = t e^ t d t+c^ aligned & y e^ g(x) =t e^ t -e^ t +c^ & y e^ g(x) =(g(x)-1) e^ g(x) +c^ & or (y-g(x)+1) e^ g(x) =c^ aligned Taking log on both sides, we get g(x)+ (1+y-g(x))=c Where, c= c^