Function of Higher Order Derivative Type 2) Problem 3 | Summary and Q&A

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April 3, 2022
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Function of Higher Order Derivative Type 2) Problem 3

TL;DR

Find the extremal of the functional y''^2 - y'^2 + x^2, which involves higher order derivatives.

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Key Insights

  • ✋ The problem involves finding the extremal of a functional with higher order derivatives.
  • 🫡 Euler's equation is used to solve the problem by finding the derivative of the functional with respect to different variables.
  • 🍉 The solution only consists of the complementary function (CF) because there is no particular integral (PI) term.
  • 😀 The final solution is y = f(x), where f(x) represents the CF.
  • 🫚 The equation simplifies to D^4y + D^2y = 0, which indicates that the roots of D squared are 0.
  • 💁 The solution includes CF terms in the form c1 + c2x + (c3cos(x) + c4sin(x)), where c1, c2, c3, and c4 are constants.
  • 🙈 The general solution is obtained by ignoring any particular integral (PI) terms because the RHS of the equation is 0.

Transcript

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Questions & Answers

Q: What is the objective of the given problem?

The objective is to find the extremal of the functional y''^2 - y'^2 + x^2.

Q: What equation is used to solve the problem?

Euler's equation, which involves finding the derivative of the functional with respect to different variables.

Q: What is the term CF in the solution?

CF stands for complementary function, which is the general solution obtained by ignoring any particular integral (PI) terms.

Q: What is the final solution of the problem?

The final solution is y = f(x), where f(x) represents the CF.

Summary & Key Takeaways

  • The problem involves finding the extremal of a functional with higher order derivatives.

  • Euler's equation is used to solve the problem.

  • By simplifying the equation, we find that the solution only consists of the complementary function (CF).

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