Can someone explain this to me please?

[AdrianZ]

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=e^1/2.

Then I substituted y(t)=Ce^x2 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?

 

[AdrianZ]

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.

 

[omkar13]

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

 

[yus310]

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=e^1/2.

Then I substituted y(t)=Ce^x2 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

 

[AdrianZ]

I guess I've already found an explanation to this.
C must be zero. probably that's the only answer this integral equation can have. other answers would lead to contradiction.