• C++ Programming for Financial Engineering
    Highly recommended by thousands of MFE students. Covers essential C++ topics with applications to financial engineering. Learn more Join!
    Python for Finance with Intro to Data Science
    Gain practical understanding of Python to read, understand, and write professional Python code for your first day on the job. Learn more Join!
    An Intuition-Based Options Primer for FE
    Ideal for entry level positions interviews and graduate studies, specializing in options trading arbitrage and options valuation models. Learn more Join!

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

(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!
 
Top