How do I rigorously prove that - two distinct solutions of a first order linear differential equation don't intersect?

[Quasar Chunawala]

(Page 125 of file, page 106, problem 19 of the book - Linear Analysis)

Hi folks,

I tried to come up with a proof for the following statement. But, I am facing trouble proceeding with the arguments. It would be nice, if someone could help. Pardon me for posting the question here, as there's no math sub-forum here.

Q. Show that two distinct solutions of a normal first order linear differential equation cannot have a point of intersection.

Proof.

Let \(L=a_{1}(x)D+a_{0}(x)\) be the linear differential operator of the first order and let
\[ Ly=h \]
be the given differential equation.

Suppose \(f(x)\) and \(g(x)\) are two distinct solutions of the linear differential equation, such that

\[Lf=h, \text{whenever }y(x_1)=k_1\]
\[Lg=h, \text{whenever }y(x_2)=k_2\]

But, I don't know where to go from here. Any tips on how to proceed would help me.

Thanks guys!

 

[Daniel Duffy]

Step 1 reduce scope

take dy/dt + ay = 0 for starters. Then generalise
BTW the statement of Q19 is a bit ambiguous..

A 1st order linear ODE has an analytic solution see page 97, eq. 3-20.
Maybe that helps.

Integrating factor - Wikipedia

en.wikipedia.org

 

[Quasar Chunawala]

Yeah, I can prove it, if I choose the analytic closed form of the solution for the first order linear ODE. Thanks Daniel.