Did I do this circuit correct If not what did I do wrongSolu

Did I do this circuit correct? If not what did I do wrong?

Solution

Ans) Whatever you did is right when Vc(0)=0,but in the problem given Vc(0)=v/2 ,so you have done wrong

With Vc(0)=v/2;

your differential equation for current will be same

in finding constant K=i(o)=(v-Vc(0))/R=v/2R you kept v/R beacuse you have not taken the Vc(0),the direction of Vc(0) is also important if is in reverse direction i(o)=(v+Vc(0))/R

Vout(t) will be simply Vout(t)=i(t)*R

your KVL equation is wrong

V(t)=Vc(t)+I(t)*R where I(t)*R=Vout(t) see the difference with your equation where you omitted Vc(t)