Finding interval where second order ODE has unique solution

In summary, the speaker is asking for help on how to apply the existence and uniqueness theorem for an nth order initial value problem. They provide an example of a second order linear ODE and mention that the theorem is verified in this case. They end their message with a closing statement and their name.
  • #1
find_the_fun
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I'm a little stuck getting started on this question. \(\displaystyle y''+\tan(x)y=e^x\) with \(\displaystyle y(0)=1,y'(0)=0\). I know the existence and uniqueness theorem
Let \(\displaystyle a_n(x),a_{n-1}(x),...,a_0(x)\) and g(x) be continuous on an Interval I and let \(\displaystyle a_n(x)\) not be 0 for every x in this interval. If x=x_0 is any point in this interval, then a solution Y(x) of the initial value problem exists on the interval and is unique.
for an nth order initial value problem

How do I apply the theorem?
 
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  • #2
find_the_fun said:
I'm a little stuck getting started on this question. \(\displaystyle y''+\tan(x)y=e^x\) with \(\displaystyle y(0)=1,y'(0)=0\). I know the existence and uniqueness theorem
for an nth order initial value problem

How do I apply the theorem?

Let's write the second order linear ODE as...

$\displaystyle \cos x\ y^{\ ''} + \sin x\ y = e^{x}\ \cos x, \ y(0)=1,\ y^{\ '} (0)=0\ (1)$

Here $a_{2}(x) = \cos x$, and is $a_{2} (0) = 1 \ne 0$, so that the existence and uniqueness theorem is verified...

Kind regards

$\chi$ $\sigma$
 

FAQ: Finding interval where second order ODE has unique solution

What is a second order ODE?

A second order ordinary differential equation (ODE) is a mathematical equation that describes the relationship between a function and its derivatives up to the second order. It is often used to model physical systems in science and engineering.

How do you find the interval where a second order ODE has a unique solution?

To find the interval where a second order ODE has a unique solution, you must first solve the equation using standard techniques such as separation of variables or the method of undetermined coefficients. Then, you can determine the interval of validity by examining the initial or boundary conditions given in the problem. The solution will be unique within this interval.

What is the importance of finding the interval where a second order ODE has a unique solution?

Knowing the interval where a second order ODE has a unique solution is important because it ensures that the solution is well-defined and accurate. Without this knowledge, the solution may not be valid or it may have multiple solutions, making it difficult to interpret and use in real-world applications.

What happens if a second order ODE does not have a unique solution?

If a second order ODE does not have a unique solution, it means that there are multiple solutions that satisfy the equation and initial/boundary conditions. This can occur when the interval of validity is not specified or when there are not enough initial/boundary conditions to uniquely determine the solution. In this case, further analysis or information is needed to narrow down the solution.

Are there any techniques or methods to determine the interval where a second order ODE has a unique solution?

Yes, there are several techniques and methods that can be used to determine the interval where a second order ODE has a unique solution. These include solving the equation and examining the initial/boundary conditions, using graphical methods to visualize the solution, or utilizing numerical methods such as Euler's method to approximate the solution within a given interval.

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