2nd order ODE solution bases /wronskain question

• Greger
In summary: The Wronskian should be equal to the product of the coefficients c1c4-c2c3, which shows that the two bases are indeed multiples of each other. In summary, the conversation discusses a question on finding the relationship between two different bases of solutions using the Wronskian. The solution is found by calculating the Wronskian of the second basis and relating it to the Wronskian of the first basis, showing that the two bases are just multiples of each other.
Greger
hello

this question from my coarse notes has been giving me some trouble so i thought i would ask for some help on here,

http://img88.imageshack.us/img88/9764/asfar.jpg

i understand that since the bases are bases of the same solutions then they are just a multiple of each other, but I'm not sure how you would show it using the wronskain.

i first tried starting by saying since both are two different bases for the solutions then
$k\phi_{1}=\psi_{1}$

$k\phi_{2}=\psi_{2}$

then doing the wronskian, but it gives k2

it seems like there is something straight forward that i am not seeing

anyone know what it might be?

Last edited by a moderator:
Greger said:
hello

this question from my coarse notes has been giving me some trouble so i thought i would ask for some help on here,

http://img88.imageshack.us/img88/9764/asfar.jpg

i understand that since the bases are bases of the same solutions then they are just a multiple of each other, but I'm not sure how you would show it using the wronskain.

No, they aren't just multiples of each other. For example, the equation ##y''-y=0## could have ##y=A\cosh(x)+B\sinh(x)## or ##y=Ce^x+De^{-x}##.
i first tried starting by saying since both are two different bases for the solutions then
$k\phi_{1}=\psi_{1}$

$k\phi_{2}=\psi_{2}$

then doing the wronskian, but it gives k2

it seems like there is something straight forward that i am not seeing

anyone know what it might be?

What you do know is that each function in the second basis is a linear combination of the functions in the first basis. Calciulate the Wronskian for the second basis and use that fact to relate it to the Wronskian in the first basis.

Last edited by a moderator:
Hey,

Sorry I had an assignment I had to work on before I started studying math again,

So I did what you said just now using

$c_1\phi_{1} +c_2\phi_{2}=\psi_{1}$

$c_3\phi_{1} +c_4\phi_{2}=\psi_{1}$

I got the required result with k = c1c4-c2c3

It seems good to me thanks!

Does it look right to you?

Yes, that looks to be correct.

1. What is a second order ODE?

A second order ordinary differential equation (ODE) is a mathematical equation that involves a function, its derivatives up to the second order, and an independent variable. It is commonly used to model physical systems in engineering and science.

2. What does "solution basis" mean in the context of second order ODEs?

A solution basis refers to a set of linearly independent solutions to a second order ODE. These solutions can be used to form a general solution for the ODE, which can be used to find specific solutions for different initial conditions.

3. What is the Wronskian of a second order ODE?

The Wronskian of a second order ODE is a determinant that can be used to determine the linear independence of a set of solutions to the ODE. It is defined as the determinant of a matrix formed by the solutions and their derivatives.

4. How is the Wronskian used to determine the linear independence of solutions?

If the Wronskian of a set of solutions is nonzero at a given point, then the solutions are linearly independent at that point. If the Wronskian is zero at a point, the solutions may still be linearly independent, but further analysis is needed to determine this. If the Wronskian is zero at all points, then the solutions are linearly dependent.

5. Can the Wronskian be used to find a particular solution to a second order ODE?

No, the Wronskian cannot be used to find a particular solution to a second order ODE. It is only used to determine the linear independence of solutions and does not provide information about specific solutions.

Replies
2
Views
676
Replies
3
Views
1K
Replies
8
Views
707
Replies
4
Views
2K
Replies
4
Views
1K
Replies
6
Views
2K
Replies
4
Views
1K
Replies
2
Views
3K
Replies
6
Views
2K
Replies
2
Views
2K