Homogenous Differential Equation Solution

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SUMMARY

The discussion focuses on solving the first-order linear differential equation \(\frac{dy}{dx} = x + y\) using substitution and integrating factor methods. The substitution \(v = x + y\) simplifies the equation to \(v' = v + 1\), which can be solved using separation of variables. The integrating factor method is also applicable, where the integrating factor is \(e^{-x}\), leading to the solution \(y = 2e^x - x - 1\). Participants emphasize understanding the underlying concepts rather than memorizing formulas.

PREREQUISITES
  • Understanding of first-order linear differential equations
  • Familiarity with substitution methods in differential equations
  • Knowledge of integrating factors and their application
  • Ability to perform integration and solve separable equations
NEXT STEPS
  • Study the method of integrating factors for linear differential equations
  • Learn about substitution techniques in solving differential equations
  • Explore the method of undetermined coefficients for non-homogeneous equations
  • Practice solving first-order linear differential equations with initial conditions
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Students studying differential equations, educators teaching calculus, and anyone seeking to enhance their problem-solving skills in mathematical analysis.

  • #31
Orodruin said:
I think that the approach itself is not very different from the variable substitution that you started with (just writing ##v = y + x + 1## rather than ##v = y+x## - you could also see it as making yet another substitution from ##v' = v + 1## to ##u = v+1##, which also gives ##u' = u##). Regardless, only a very ignorant teacher would deduct points for you using a working method unless explicitly asking for something to be solved with a particular method. But of course this is completely up to you.
thats true...to my knowledge the test doesn't explicitly ask for a certain method so thanks for help :)
 
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  • #32
_N3WTON_ said:
Thank you...your insight has been very helpful to me :)
You are welcome:)

ehild
 

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