- #1
gsingh2011
- 115
- 1
My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)[itex]\int[/itex]I(x)Q(x)dx where I(x) is the integrating factor.
Now the book showed how the Integrating Factor Method was developed, but it doesn't prove this theorem, particularly why a unique solution exists and why there are no other solutions of a different form (singular solutions).
Also it states, "The appropriate value of the constant C can be selected "automatically" by writing, I(x)=exp([itex]\int[/itex][itex]^{x_{0}}_{x}[/itex]P(t)dt) and y(x)=1/I(x)[y0+[itex]\int[/itex][itex]^{x_{0}}_{x}[/itex]I(t)Q(t)dt]"
I don't understand how they got this form. If you have a definite integral there shouldn't be constant like y0 either...
Now the book showed how the Integrating Factor Method was developed, but it doesn't prove this theorem, particularly why a unique solution exists and why there are no other solutions of a different form (singular solutions).
Also it states, "The appropriate value of the constant C can be selected "automatically" by writing, I(x)=exp([itex]\int[/itex][itex]^{x_{0}}_{x}[/itex]P(t)dt) and y(x)=1/I(x)[y0+[itex]\int[/itex][itex]^{x_{0}}_{x}[/itex]I(t)Q(t)dt]"
I don't understand how they got this form. If you have a definite integral there shouldn't be constant like y0 either...