2nd basis function for 2nd order ODE

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SUMMARY

The discussion focuses on finding the second linearly independent solution for the second-order ordinary differential equation (ODE) given the first solution y_1(t) = t. The user employs Abel's theorem to derive the second solution, which involves a complex integration process. Ultimately, the second solution is determined to be y_2(t) = exp(t), confirming that the two solutions are y_1(t) = t and y_2(t) = exp(t).

PREREQUISITES
  • Understanding of second-order ordinary differential equations (ODEs)
  • Familiarity with Abel's theorem in the context of ODEs
  • Knowledge of integration techniques, particularly involving exponential functions
  • Basic concepts of linear independence in the context of differential equations
NEXT STEPS
  • Study the application of Abel's theorem in solving second-order ODEs
  • Practice integration techniques involving exponential functions and telescoping series
  • Explore the theory of linear independence for solutions of differential equations
  • Learn about other methods for finding solutions to second-order ODEs, such as reduction of order
USEFUL FOR

Mathematicians, physics students, and engineers who are solving second-order ordinary differential equations and require a deeper understanding of solution methods and linear independence.

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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!
 
Physics news on Phys.org
found it, integral was a telescoping series from parts. the solution is \exp(-t)
 
Obviously, the second solution is exp(t)
 

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