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.

Transcript

hello friends in this video we'll be discussing calculus of variation type number two function of higher order derivatives problem number three welcome back friends let's have a look on the given problem here find the extremal of this is the given functional which is y double dash square minus y dash square plus X square if you see this problem we ... Read More

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.

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).