Integrating factor for first order linear equations uniqueness theorem

[gsingh2011]

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)\intI(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(\int^{x_{0}}_{x}P(t)dt) and y(x)=1/I(x)[y₀+\int^{x_{0}}_{x}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...

 

[Hootenanny]

As far as the proof of existence an uniqueness goes, it is straightforward and can be found at: http://arapaho.nsuok.edu/~okar-maa/news/okarproceedings/OKAR-2006/finan.pdf

As for the definite form of the integrating factor, you are indeed correct, no constant arises from the definite integral and therefore we must add an appropriate constant so that the solution satisfies the boundary conditions.

We have

y(x) = \frac{1}{I(x)}\left(y_0 + \int_x^{x_0} I(t)Q(t)\;\text{d}t\right)

I(x) = \exp\left(\int_x^{x_0} P(t)\;\text{d}t\right)

So, if we consider x=x_0 we findy(x_0) = \frac{1}{I(x_0)}\left(y_0 + \int_{x_0}^{x_0} I(t)Q(t)\;\text{d}t\right)

Obviously the integral vanishes, leaving

y(x_0) = \frac{y_0}{I(x_0)}

with

I(x_0) = \exp\left(\int_{x_0}^{x_0} P(t)\;\text{d}t\right) = \exp(0) = 1

Hence

y(x_0) = y_0

as required.

 

[gsingh2011]

Thanks, that makes sense. One thing I don't understand in the link you gave me is on page four, they state w(t)=y₁(t₀)-y₂(t₀)=y₀-y₀. Why are the intial values of those two functions the same, if we assumed that the two functions would be different?

 

[Hootenanny]

Thanks, that makes sense. One thing I don't understand in the link you gave me is on page four, they state w(t)=y₁(t₀)-y₂(t₀)=y₀-y₀. Why are the intial values of those two functions the same, if we assumed that the two functions would be different?

Because even though they may be different functions, if they are solutions of the same boundary value problem, then they must satisfy the same boundary conditions!

Does that make sense?