Solution of Higher Order Differential Equation Problem 9

TL;DR

Analyzing a calculus of variation problem to find the extremal of a given functional.

Transcript

hello friends in this video we'll be discussing calculus of variation type number one problem number nine welcome back friends let's have a look on the given problem here we need to find the extremal of this is the functional given in the problem which is y dash square minus y square plus two y sine x if you observe this functional you will find th... Read More

Key Insights

• ❓ The problem involves finding the extremal of a functional with multiple variables.
• ❓ Euler's equation is used to solve the problem, with two options available.
• ❓ The solution includes both the complementary function (CF) and the particular integral (PI).

Questions & Answers

Q: What is the functional given in the problem and what variables does it include?

The functional is y' square minus y square plus 2y sine x. It includes the variables x, y, and y'.

Q: How many options are there to solve the problem using Euler's equation and which option is selected?

There are two options, but the simpler one is selected for solving the problem.

Q: What is the equation obtained after applying Euler's equation?

The equation obtained is -2y plus 2 sine x minus 2 times the second derivative of y with respect to x equals minus y plus sine x.

Q: What is the solution for finding the CF and the PI?

The solution is CF + PI, where CF is c1 cos x + c2 sine x and the PI is -x/2 cos x.

Summary & Key Takeaways

• The problem involves finding the extremal of a functional that includes variables x, y, and y'.

• Using Euler's equation, the simpler option is selected to solve the problem.

• The solution includes both the complementary function (CF) and the particular integral (PI).