# Can someone explain this to me please?

In summary, the equation suggests that y(1) ≠ 0, and the y that is obtained from the ODE solution tells me that y(1)≠0. If you keep it in terms of t, you get the same function, but t^2, x^2."f

I'm asking this integral equation (I'm not sure if it's an integral equation or not by it's a problem in my ODE book and because it has an integral in it I called it that way). anyways, this is the problem:

$$y=\int^{x}_{1}ty(t)dt$$

I differentiated y with respect to x and I turned that equation into this ODE: $$y'=xy$$
Solving this ODE yields $$y=Ce^{x^2/2}$$

But from the definition of y, it is clear that y(1)=0 while my solution suggests that y=e1/2.

Then I substituted y(t)=Cex2 in the original equation and I obtained:
$$y=\int^{x}_{1}tCe^{t^2/2}dt → y=C(e^{t^2/2})|^{x}_{1}→y=C(e^{x^2/}-e^{1/2})$$
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?

Any ideas on this matter will be appreciated. Are my questions really that hard that they usually get no responses back on physics forum or there's a conspiracy against me? lol.

The condition y(1)=0 is given in order to find constant.THE PURPOSE OF GIVING BOUNDARY CONDITIONS IS TO OBTAIN CONSTANTS AFTER INTEGRATION.

I'm asking this integral equation (I'm not sure if it's an integral equation or not by it's a problem in my ODE book and because it has an integral in it I called it that way). anyways, this is the problem:

$$y=\int^{x}_{1}ty(t)dt$$

I differentiated y with respect to x and I turned that equation into this ODE: $$y'=xy$$
Solving this ODE yields $$y=Ce^{x^2/2}$$

But from the definition of y, it is clear that y(1)=0 while my solution suggests that y=e1/2.

Then I substituted y(t)=Cex2 in the original equation and I obtained:
$$y=\int^{x}_{1}tCe^{t^2/2}dt → y=C(e^{t^2/2})|^{x}_{1}→y=C(e^{x^2/}-e^{1/2})$$
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?

"In order to solve this you need to know what value of t... x and 1 correspond to and then you can go from there. If you keep it in terms of t, you get the same function, but t^2, x^2.

Knowing that x=t, at y=?, and t=0, y=?, you can transform accordingly .

This an ODe, you you don't have what y=, when x=t, t=0, etc.

YS

I guess I've already found an explanation to this.