Question

Assume that \(p\) and \(q\) are continuous, and that the functions \(y_{1}\) and \(y_{2}\) are solutions of the differential equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) on an open interval \(I\) Prove that if \(y_{1}\) and \(y_{2}\) are zero at the same point in \(I,\) then they cannot be a fundamental set of solutions on that interval.

Short answer

Question: Prove that if y1 and y2 are zero at the same point within the open interval I, then they cannot form a fundamental set of solutions for a given second-order differential equation on that interval. Answer: Since the Wronskian at point \(t_0\) is zero, it means that the two solutions \(y_{1}\) and \(y_{2}\) are linearly dependent. Hence, they cannot form a fundamental set of solutions on the open interval \(I\).

Step-by-step solution

Write down the given differential equation and solutions

We have the differential equation given as: \(y^{\prime \prime} + p(t)y^{\prime} + q(t)y = 0\) with solutions \(y_{1}(t)\) and \(y_{2}(t)\).

Define the Wronskian

The Wronskian W is a measure of the linear independence of the two solutions. It is defined as: \(W(y_{1}, y_{2})(t) = y_{1}(t)y_{2}^{\prime}(t) - y_{1}^{\prime}(t)y_{2}(t)\)

Establish the given condition and compute the Wronskian

We are given the condition that both \(y_{1}\) and \(y_{2}\) are zero at the same point in the open interval \(I\). Let's call this point \(t_0\). So, \(y_{1}(t_0) = y_{2}(t_0) = 0\) Now, let's calculate the Wronskian at this point, \(t_0\): \(W(y_{1}, y_{2})(t_0) = y_{1}(t_0)y_{2}^{\prime}(t_0) - y_{1}^{\prime}(t_0)y_{2}(t_0)\) As both \(y_{1}(t_0)\) and \(y_{2}(t_0)\) are zero, the Wronskian at this point becomes: \(W(y_{1}, y_{2})(t_0) = 0·y_{2}^{\prime}(t_0) - y_{1}^{\prime}(t_0)·0 = 0\)

Conclude that y1 and y2 do not form a fundamental set of solutions

Since the Wronskian at point \(t_0\) is zero, it means that the two solutions \(y_{1}\) and \(y_{2}\) are linearly dependent. Hence, they cannot form a fundamental set of solutions on the open interval \(I\).

Key concepts

Wronskian

Imagine you're given two different solutions to a music puzzle—the Wronskian tells you if the solutions really strike different notes or if they're just echoes of each other! In the world of differential equations, understanding if two solutions are unique is crucial. That's where the Wronskian comes into play.

The Wronskian, named after the mathematician Józef Hoene-Wroński, is like a special test. If you have two functions, say, two answers to a tricky question, the Wronskian helps you see if those answers truly are different, or if they somehow depend on each other. If the Wronskian is not zero, then you can be sure your solutions are like two distinct instruments in an orchestra—each playing its own part. But if it turns out to be zero, it's like discovering that both instruments were playing the same tune all along—so you've really only got one solution, not two.

For example, suppose you have two functions, \(y_1\) and \(y_2\), you calculate the Wronskian using this magic formula: \[W(y_{1}, y_{2})(t) = y_{1}(t)y_{2}^{\prime}(t) - y_{1}^{\prime}(t)y_{2}(t)\] When it equals zero at any point, it's telling you that your two functions are just playing back the same old song, not offering anything new.

Linear Independence

Now, let's chat about what it means for two things to be linearly independent—it's like having two super spies that can pull off their missions without ever crossing paths. In the realm of differential equations, two functions are linearly independent if no amount of clever multiplying and adding them together can make them mimic each other.

Here's the catch: for linear independence, we need to make sure our functions aren't just different at one spot, but everywhere along the journey. That means, no matter how you mix them, one function can't be written as a multiple of the other. Just like having two unique spices in a recipe, each adds its own distinct flavor that the other can't replace.

Why is this so important? Because in differential equations, only when functions are linearly independent do they bring new information to the table, allowing us to solve really complex problems. If they're not, then maybe we're missing out on some crucial spice that makes our solution complete.

Fundamental Set of Solutions

Imagine you're assembling the ultimate toolkit, where every tool has a special role to play—this is the idea behind a fundamental set of solutions in differential equations. It's that perfect collection where each solution is crucial, and altogether, they can tackle any problem the equation throws at you. This set forms the 'basis' for all possible solutions.

But beware—a fundamental set of solutions only works if each solution is linearly independent from the others. Think of it like a group of superheroes, where each has a unique superpower. If two heroes have the same power, they're not really contributing anything different to the team.

So for a set of solutions to be truly fundamental, you need that each member stands out on its own, much like having a set of keys where no two keys can open the same lock. In the language of differential equations, that means a fundamental set allows us to express any solution to the equation as a combination of the solutions from our all-star team.

Second-order Linear Differential Equation

Picture a staircase with a twist where each step depends on the previous two; that's similar to how a second-order linear differential equation operates. These equations are the foundation of many natural phenomena, from the vibrations of a guitar string to the orbits of planets.

A second-order linear differential equation looks something like this: \[y^{\prime \prime} + p(t)y^{\prime} + q(t)y = 0\] In this show-stopping formula, \(y^{\prime \prime}\) represents the second derivative of \(y\), which is like the acceleration if \(y\) were the position—it tells you how quickly the rate of change is changing! The functions \(p(t)\) and \(q(t)\) choreograph how the steps (terms) of the equation play along with \(y\).

What's super cool about these equations is that when you find two linearly independent solutions, you've got yourself a dynamic duo that describes every performance the equation can put on. They're your fundamental set, leading you to uncover the mysteries of how things move and interact in a symphony of calculus.