Finding the Homogenous Solution to a Variable Coefficients 2nd order ODE

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SUMMARY

The discussion centers on solving the homogeneous solution of the variable coefficients second-order ordinary differential equation (ODE) given by x y'' + (x + 1) y' = 2x. The key method identified for finding the complementary function is separation of variables, specifically transforming the equation into a first-order ODE by letting y' = z. This approach allows for the application of established techniques to derive the solution effectively.

PREREQUISITES
  • Understanding of second-order ordinary differential equations
  • Familiarity with separation of variables technique
  • Knowledge of first-order ODEs
  • Basic concepts of complementary functions in differential equations
NEXT STEPS
  • Study the method of separation of variables in detail
  • Learn about complementary functions and their applications in ODEs
  • Explore the variation of parameters method for non-homogeneous equations
  • Investigate ansatz methods for solving differential equations
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Mathematicians, engineering students, and anyone involved in solving differential equations, particularly those dealing with variable coefficients and seeking to understand advanced solution techniques.

joelio36
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x y'' + (x + 1) y' = 2 x

Solve for y(x).

Due to the coefficients being a function of x, I have no idea where to start to find the homogenous solution (Complementary Function). I know how to proceed after this part with the variation of parameters method.

I just have no idea where to begin to find a solution of this equation, which seems to be the pre-requisite for all solution methods! Do you just used ansatz? If so how would I pick an ansatz?

I'm tearing my hair out here, any help would be greatly appreciated.
 
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The homogenous equation x y'' + (x + 1) y' = 0 can be solved by separation of variables.
 
y' = z
then separation of variables in a first order ODE.
 

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