Solve ODE with direct integration

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SUMMARY

The discussion centers on solving ordinary differential equations (ODEs) through direct integration, specifically addressing the challenges of integrating functions with terms like (x - x0) and (t - tau). The participant struggles with the concept of dummy variables in integration and the derivative of an integral involving both dummy and real variables. A key takeaway is the application of the Cauchy formula for repeated integration, which simplifies the process by allowing the expression of repeated integrals as single integrals.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with integration techniques, particularly integration by parts
  • Knowledge of dummy variables in calculus
  • Basic principles of derivatives and integrals
NEXT STEPS
  • Study the Cauchy formula for repeated integration in detail
  • Learn about integration by parts and its applications in solving ODEs
  • Explore the concept of dummy variables and their role in calculus
  • Practice solving ODEs using direct integration techniques
USEFUL FOR

Students studying calculus, mathematicians focusing on differential equations, and educators teaching integration techniques.

veneficus5
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Homework Statement


Latex takes me forever so I'm going to take a picture



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The Attempt at a Solution



I'm having issues with integrating functions. There seems to be this (x-x0) term that crops up everywhere. Last time it was (t - tau). It's always (variable - dummy variable) so far, but I don't understand why. I also tried plugging it into the original, but I didn't know how to take the derivative of an integral with a mix of dummy and real variables. I guess this is something I should have learned in calculus but never did?

Thanks a lot!
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Recall from elementary calculus the Cauchy formula for repeated integration, which follows from integrationby parts and allows one to write a repeated integral as a single integral. In other words you have found two equivelant ways to express the answer.
 

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