Solution of Higher Order Differential Equation Problem 11

TL;DR

Solving a calculus of variation problem involving the integration of a function with y, x, and y' terms.

Transcript

hello friends in this video we'll be discussing calculus of variation type number one problem number 11 welcome back friends let's have a look on the given problem here we need to integrate between x 1 to x 2 to y square plus y - square plus 2x cubed y DX this is the functional given in the problem who y square plus y dash square plus 2x cubed y if... Read More

Key Insights

• ❣️ The problem involves solving a calculus of variation problem with y, x, and y' terms.
• 🌆 Euler's equation is used to find the derivatives with respect to y and y', which are set to zero.
• ❓ The CF + PI method is employed to solve the problem, where CF is the complementary function and PI is the particular integral.

Questions & Answers

Q: What is the main objective of this calculus of variation problem?

The main objective is to find the integral of the given function y square plus y - square plus 2x cubed y.

Q: How is Euler's equation used in solving this problem?

Euler's equation states that F upon dou Y minus D by DX of dou F upon dou y' should be equal to zero. By substituting the function in Euler's equation, the derivatives are calculated.

Q: What is the auxiliary equation in this problem?

The auxiliary equation is given by d square minus 2 equal to 0, where d is the differential operator.

Q: How is the particular integral (PI) determined in this problem?

The particular integral is determined by expanding the RHS of the auxiliary equation, which is X cube, using expansion formulas for 1 minus X inverse.

Summary & Key Takeaways

• The problem requires finding the integral of the function y square plus y - square plus 2x cubed y.

• By applying Euler's equation, the derivative of the function with respect to y and y' is determined to be zero.

• The problem is then solved using the CF + PI method, where CF is the complementary function and PI is the particular integral.