Cauchy Method/ UC set OR Variation of Param?

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SUMMARY

The discussion centers on solving a differential equation using the Cauchy method and the method of variation of parameters. The equation in question is i=primes (x^2)(y^ii)+(x)(y^i)+y=4sin(lnx). The user encounters discrepancies in results when applying the Undetermined Coefficients method and variation of parameters, particularly noting that the Undetermined Coefficients method is not applicable for functions like tan(x), sec(x), and ln(x). The substitution u=ln(x) simplifies the problem to a differential equation with constant coefficients, y'' + y = ln(u).

PREREQUISITES
  • Understanding of differential equations, specifically second-order linear equations.
  • Familiarity with the Cauchy method for solving differential equations.
  • Knowledge of the method of variation of parameters.
  • Experience with the Undetermined Coefficients method in differential equations.
NEXT STEPS
  • Study the application of the Cauchy method in solving differential equations.
  • Learn about the method of variation of parameters in detail.
  • Investigate the limitations of the Undetermined Coefficients method, especially with specific functions.
  • Explore the implications of substituting u=ln(x) in differential equations with constant coefficients.
USEFUL FOR

Students and professionals in mathematics, particularly those focused on differential equations, as well as educators seeking to clarify methods for solving complex equations.

dgutierrez079
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Hello new to this forum , Was solving some Diff eq problems and iam getting two different answers using two methods, ok the problem is i=primes (x^2)(y^ii)+(x)(y^i)+y=4sin(lnx)
This is cauchy method, When i use variation of parameters i get a long answer with impossible integrals and when i use Undetermined coefficients i also get an answer, Not sure witch is correct i also understand that Uc method can't be used with tanx, secx, lnx, but in this case that there is a sin in there iam not sure. Thanks
 
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Please show what you did and what answer you got.

(Note that the substution u= ln(x) turns this into the differential equation with constant coefficients, y''+ y= ln(u) where the primes now indicate differentiation with respect to u.)
 

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