Solve the integral equation $$y(x) = 3 + 2\int_1^x t \ y(t) \ dt $$
First I solved for the integral equation. Then I'm told to differentiate and I get
$${dy \over dx} = 2 x y(x) $$
Then I see that they're separable and I use that so I take both of the integrals after arranging the functions. What I get is
$$ \ln(y(x))=x^2+C$$
This I don't get, however: $$y(x) = C_1 \exp(x^2)$$
Where does this come from?
Now I choose an initial value for the first equation and I choose $ x=1$
$$ y(1) = 3 + 0 = 3$$
Now here's the problem
After this it's supposed to be $C_1= 3/e$
and the solution is
$$ y(x) = 3 \exp(x^2-1)$$
I understand everything up til choosing the initial value but i'm guessing it might be arbitrary however what happens with the $C_1$? How is this calculated? Everything gets really confusing here.