2nd order differential equations

In summary, if p and q are continuous on an open interval I and y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on I, then they cannot have a common inflection point in I unless p and q are both 0 at that point. Additionally, if 0 E I, then y(t)=t^3 cannot be a solution of the differential equation on I, as it contradicts the existence-uniqueness theorem.
  • #1
kingwinner
1,270
0
1) Assume that p and q are continuous on some open interval I, and that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on I.
a) Prove that if {y1, y2} form a fundamental set of solutions on I, then they can't have a common inflection point in I, unless p and q are both 0 at this point.
b) If 0 E I, prove that y(t)=t3 cannot be a solution of the differential equation on I.


=====================
1a) My first question: "A unless B", is this equivalent to "A if and only if not B". That is, do I have to prove both directions for question 1a simply by seeing the word "unless"?

Now my thoughts about this problem:
Inflection point at c => f ''(c)=0 or f ''(c) does not exist.
But in this case, since we're given that y1 and y2 are solutions of the differential equation, this means that y1 and y2 must be twice differentiable, so we can reject the possiblity f ''(c) does not exist.
Now we have: inflection point at c => f ''(c)=0
Fundamental set of solutions means Wronskian is nonzero.
But now I am having a lot of trouble deriving one from the another...


1b) No clue...need some hints...


Thanks for your help!:smile:
 
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  • #2
For part b, since we are given that p, q are continuous, I think this may be related to existence-uniqueness theorem...but I can't figure out how to use it...
 
  • #3
1b) y(t)=t^3
=>y'(t)=3t^2
=>y''(t)=6t

=>y'(0)=0, y''(0)=0

But the initial value problem (IVP) y'' + p(t)y' + q(t)y = 0, y'(0)=0, y''(0)=0 has the solution y≡0 by inspection, and by existence-uniqueness theorem, y≡0 must be the only solution to the IVP. So y(t)=t^3 cannot be a solution to the ODE on I if 0 E I.

I've tried my best. Is this a correct proof?
 

What is a 2nd order differential equation?

A 2nd order differential equation is a mathematical equation that involves a function and its first and second derivatives. It is commonly used to model physical systems in engineering and science.

What are the types of 2nd order differential equations?

The two main types of 2nd order differential equations are ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve only one independent variable, while PDEs involve multiple independent variables.

What are the methods for solving 2nd order differential equations?

Some common methods for solving 2nd order differential equations include separation of variables, substitution, power series, and numerical methods such as Euler's method or the Runge-Kutta method.

How are 2nd order differential equations used in science and engineering?

2nd order differential equations are used to model and analyze physical systems in various fields such as physics, chemistry, biology, and engineering. They can help us understand the behavior of complex systems and make predictions about their future behavior.

What are the applications of 2nd order differential equations?

2nd order differential equations have many applications in real-world problems such as population growth, motion of objects, heat transfer, and electrical circuits. They are also used in fields such as control systems, signal processing, and image processing.

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