2nd basis function for 2nd order ODE

In summary, a 2nd basis function for 2nd order ODE is a mathematical function used to represent the solution to a second order ordinary differential equation (ODE). It differs from a 1st basis function in that it represents the solution to a second order ODE and is composed of two linearly independent solutions. The purpose of using a 2nd basis function is to simplify the process of solving the ODE. The 2nd basis function for a specific ODE can be determined by finding two linearly independent solutions and constructing the function as a linear combination of these solutions. However, it can only be used to solve linear second order ODEs, not nonlinear ones.
  • #1
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154
2
i have the first solution y_1(t) = t for (1-t)y'' + ty' - y = 0.

I need to get the 2nd linearly independent using Abels theorem.

the integration is messy but i have it set up (sorry no latex);

y_2 = (t) * integral to t ( 1/s^2 * exp( -integral to t (s(s+1) ds) ) ds.

Could anyone show me how to do this integration step by step?

Thanks in advance!
 
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  • #2
found it, integral was a telescoping series from parts. the solution is \exp(-t)
 
  • #3
Obviously, the second solution is exp(t)
 

FAQ: 2nd basis function for 2nd order ODE

What is a 2nd basis function for 2nd order ODE?

A 2nd basis function for 2nd order ODE is a mathematical function used to represent the solution to a second order ordinary differential equation (ODE). It is typically expressed as a linear combination of two linearly independent solutions to the ODE.

How is a 2nd basis function for 2nd order ODE different from a 1st basis function?

A 2nd basis function for 2nd order ODE is different from a 1st basis function in that it represents the solution to a second order ODE, while a 1st basis function represents the solution to a first order ODE. Additionally, a 2nd basis function is composed of two linearly independent solutions, while a 1st basis function only has one solution.

What is the purpose of using a 2nd basis function for 2nd order ODE?

The purpose of using a 2nd basis function for 2nd order ODE is to simplify the process of solving the ODE. By expressing the solution as a linear combination of two linearly independent solutions, the ODE can be solved using standard methods such as separation of variables or the method of undetermined coefficients.

How do you determine the 2nd basis function for a specific 2nd order ODE?

The 2nd basis function for a specific 2nd order ODE can be determined by first finding the two linearly independent solutions to the ODE. These solutions can then be used to construct the 2nd basis function, which is typically expressed as a linear combination of the two solutions.

Can a 2nd basis function for 2nd order ODE be used to solve any 2nd order ODE?

No, a 2nd basis function for 2nd order ODE can only be used to solve linear second order ODEs. Nonlinear second order ODEs require different methods of solution and cannot be solved using a 2nd basis function.

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