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[tex]y=\int^{x}_{1}ty(t)dt[/tex]

I differentiated y with respect to x and I turned that equation into this ODE: [tex]y'=xy[/tex]

Solving this ODE yields [tex]y=Ce^{x^2/2}[/tex]

But from the definition of y, it is clear that y(1)=0 while my solution suggests that y=e

^{1/2}.

Then I substituted y(t)=Ce

^{x2}in the original equation and I obtained:

[tex]y=\int^{x}_{1}tCe^{t^2/2}dt → y=C(e^{t^2/2})|^{x}_{1}→y=C(e^{x^2/}-e^{1/2})[/tex]

And in this case y(1) is indeed equal to 0.

Would someone explain why the y that is obtained from the ODE solution tells me that y(1)≠0? What's wrong in my solution?